Find lim_(x rarr-2)(3-sqrt(6x+21))/(x+2). Choose 1 answer: (A) -1 (B) -2 (C) -3 (D) The limit doesn't exist

Substitute x = -2: Plug in the value of x = -2 directly into the function to see if the limit can be evaluated by substitution. lim_(x -> -2)(3 - sqrt(6x + 21)) / (x + 2) = (3 - sqrt(6(-2) + 21)) / (-2 + 2)Calculate square root: Calculate the value inside the square root: 6(-2) + 21 = -12 + 21 = 9.Factor numerator: Calculate the square root of 9, which is 3.Simplify expression: Substitute the square root value back into the function: (3 - 3) / (0).Final value: Simplify the numerator: 3 - 3 = 0.We have 0 / 0, which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. Multiply the numerator and denominator by the conjugate of the numerator: (3 + sqrt(6x + 21)).Apply the difference of squares formula to the numerator: (a - b)(a + b) = a^2 - b^2. (3 - sqrt(6x + 21))(3 + sqrt(6x + 21)) = 3^2 - (sqrt(6x + 21))^2Calculate the squares: 3^2 = 9 and (sqrt(6x + 21))^2 = 6x + 21.Substitute the squares back into the expression: 9 - (6x + 21).Simplify the expression: 9 - 6x - 21.Combine like terms: -6x - 12.Now we have the simplified expression: (-6x - 12) / ((x + 2)(3 + sqrt(6x + 21))).Factor out a -6 from the numerator: -6(x + 2).Cancel out the (x + 2) terms in the numerator and denominator.We are left with -6 / (3 + sqrt(6x + 21)).Now plug in x = -2 into the simplified expression: -6 / (3 + sqrt(6(-2) + 21)).Calculate the value inside the square root again: 6(-2) + 21 = 9.Calculate the square root of 9, which is 3, again.Substitute the square root value back into the simplified expression: -6 / (3 + 3).Simplify the denominator: 3 + 3 = 6.Calculate the final value of the limit: -6 / 6.Simplify the fraction: -6 / 6 = -1.

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Find
lim_(x rarr-2)(3-sqrt(6x+21))/(x+2).
Choose 1 answer:
(A) -1
(B) -2
(C) -3
(D) The limit doesn't exist

Find $\lim _{x \rightarrow-2} \frac{3-\sqrt{6 x+21}}{x+2}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-1$ $\newline$ (B) $-2$ $\newline$ (C) $-3$ $\newline$ (D) The limit doesn't exist

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Algebra 1

Domain and range of square root functions: equations

Full solution

Q. Find

\lim _{x \rightarrow-2} \frac{3-\sqrt{6 x+21}}{x+2}

\newline

Choose

1

answer:

\newline

(A)

-1

\newline

(B)

-2

\newline

(C)

-3

\newline

(D) The limit doesn't exist

Substitute $x = -2$ : Plug in the value of $x = -2$ directly into the function to see if the limit can be evaluated by substitution. $\newline$ $\lim_{x \to -2}\frac{3 - \sqrt{6x + 21}}{x + 2} = \frac{3 - \sqrt{6(-2) + 21}}{-2 + 2}$
Calculate square root: Calculate the value inside the square root: $6(-2) + 21 = -12 + 21 = 9$ .
Factor numerator: Calculate the square root of $9$ , which is $3$ .
Simplify expression: Substitute the square root value back into the function: $(3 - 3) / (0)$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $\frac{0}{0}$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $\frac{0}{0}$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $\frac{0}{0}$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .Calculate the value inside the square root again: $0 / 0$ $7$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .Calculate the value inside the square root again: $0 / 0$ $7$ .Calculate the square root of $0 / 0$ $8$ , which is $0 / 0$ $9$ , again.
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .Calculate the value inside the square root again: $0 / 0$ $7$ .Calculate the square root of $0 / 0$ $8$ , which is $0 / 0$ $9$ , again.Substitute the square root value back into the simplified expression: $(3 + \sqrt{6x + 21})$ $0$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .Calculate the value inside the square root again: $0 / 0$ $7$ .Calculate the square root of $0 / 0$ $8$ , which is $0 / 0$ $9$ , again.Substitute the square root value back into the simplified expression: $(3 + \sqrt{6x + 21})$ $0$ .Simplify the denominator: $(3 + \sqrt{6x + 21})$ $1$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .Calculate the value inside the square root again: $0 / 0$ $7$ .Calculate the square root of $0 / 0$ $8$ , which is $0 / 0$ $9$ , again.Substitute the square root value back into the simplified expression: $(3 + \sqrt{6x + 21})$ $0$ .Simplify the denominator: $(3 + \sqrt{6x + 21})$ $1$ .Calculate the final value of the limit: $(3 + \sqrt{6x + 21})$ $2$ .
Final value: Simplify the numerator: $3 - 3 = 0$ .We have $0 / 0$ , which is an indeterminate form, so we cannot determine the limit by direct substitution.Factor the numerator using the conjugate to simplify the expression. $\newline$ Multiply the numerator and denominator by the conjugate of the numerator: $(3 + \sqrt{6x + 21})$ .Apply the difference of squares formula to the numerator: $(a - b)(a + b) = a^2 - b^2$ . $\newline$ $(3 - \sqrt{6x + 21})(3 + \sqrt{6x + 21}) = 3^2 - (\sqrt{6x + 21})^2$ Calculate the squares: $3^2 = 9$ and $(\sqrt{6x + 21})^2 = 6x + 21$ .Substitute the squares back into the expression: $9 - (6x + 21)$ .Simplify the expression: $9 - 6x - 21$ .Combine like terms: $-6x - 12$ .Now we have the simplified expression: $0 / 0$ $0$ .Factor out a $0 / 0$ $1$ from the numerator: $0 / 0$ $2$ .Cancel out the $0 / 0$ $3$ terms in the numerator and denominator.We are left with $0 / 0$ $4$ .Now plug in $0 / 0$ $5$ into the simplified expression: $0 / 0$ $6$ .Calculate the value inside the square root again: $0 / 0$ $7$ .Calculate the square root of $0 / 0$ $8$ , which is $0 / 0$ $9$ , again.Substitute the square root value back into the simplified expression: $(3 + \sqrt{6x + 21})$ $0$ .Simplify the denominator: $(3 + \sqrt{6x + 21})$ $1$ .Calculate the final value of the limit: $(3 + \sqrt{6x + 21})$ $2$ .Simplify the fraction: $(3 + \sqrt{6x + 21})$ $3$ .

Find $\lim _{x \rightarrow-2} \frac{3-\sqrt{6 x+21}}{x+2}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-1$ $\newline$ (B) $-2$ $\newline$ (C) $-3$ $\newline$ (D) The limit doesn't exist

Full solution

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Find $\lim _{x \rightarrow-2} \frac{3-\sqrt{6 x+21}}{x+2}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-1$ $\newline$ (B) $-2$ $\newline$ (C) $-3$ $\newline$ (D) The limit doesn't exist