Find lim_(x rarr4)(2-sqrt(4x-12))/(x-4). Choose 1 answer: (A) 2 (B) 1 (C) -1 (D) The limit doesn't exist

Check Indeterminate Form: Substitute the value of x approaching 4 into the limit expression to see if it results in an indeterminate form. \[ \lim_{x \to 4} \frac{2 - \sqrt{4x - 12}}{x - 4} \] Substituting x = 4 gives us: \[ \frac{2 - \sqrt{4(4) - 12}}{4 - 4} = \frac{2 - \sqrt{16 - 12}}{0} = \frac{2 - 2}{0} = \frac{0}{0} \] This is an indeterminate form, so we need to use algebraic manipulation to simplify the expression.Multiply by Conjugate: Multiply the numerator and denominator by the conjugate of the numerator to rationalize the expression. The conjugate of 2 - \sqrt{4x - 12} is 2 + \sqrt{4x - 12}. We multiply the numerator and denominator by this conjugate: \[ \lim_{x \to 4} \frac{(2 - \sqrt{4x - 12})(2 + \sqrt{4x - 12})}{(x - 4)(2 + \sqrt{4x - 12})} \]Apply Difference of Squares: Apply the difference of squares formula to the numerator. \[ (2 - \sqrt{4x - 12})(2 + \sqrt{4x - 12}) = 2^2 - (\sqrt{4x - 12})^2 = 4 - (4x - 12) \] Simplify the numerator: \[ 4 - 4x + 12 = 16 - 4x \] Now the limit expression is: \[ \lim_{x \to 4} \frac{16 - 4x}{(x - 4)(2 + \sqrt{4x - 12})} \]Factor Out -4: Factor out -4 from the numerator to cancel out the (x - 4) term in the denominator. \[ \lim_{x \to 4} \frac{-4(x - 4)}{(x - 4)(2 + \sqrt{4x - 12})} \] Now we can cancel out the (x - 4) terms: \[ \lim_{x \to 4} \frac{-4}{2 + \sqrt{4x - 12}} \]Substitute and Simplify: Substitute x = 4 into the simplified limit expression. \[ \lim_{x \to 4} \frac{-4}{2 + \sqrt{4x - 12}} = \frac{-4}{2 + \sqrt{4(4) - 12}} = \frac{-4}{2 + \sqrt{16 - 12}} = \frac{-4}{2 + 2} = \frac{-4}{4} = -1 \]

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Find
lim_(x rarr4)(2-sqrt(4x-12))/(x-4).
Choose 1 answer:
(A) 2
(B) 1
(C) -1
(D) The limit doesn't exist

Find $\lim _{x \rightarrow 4} \frac{2-\sqrt{4 x-12}}{x-4}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $2$ $\newline$ (B) $1$ $\newline$ (C) $-1$ $\newline$ (D) The limit doesn't exist

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Math Problems

Algebra 2

Solve quadratic inequalities

Full solution

Q. Find

\lim _{x \rightarrow 4} \frac{2-\sqrt{4 x-12}}{x-4}

\newline

Choose

1

answer:

\newline

(A)

2

\newline

(B)

1

\newline

(C)

-1

\newline

(D) The limit doesn't exist

Check Indeterminate Form: Substitute the value of x approaching $4$ into the limit expression to see if it results in an indeterminate form. $\newline$ $\lim_{x \to 4} \frac{2 - \sqrt{4x - 12}}{x - 4}$ $\newline$ Substituting x = $4$ gives us: $\newline$ $\frac{2 - \sqrt{4(4) - 12}}{4 - 4} = \frac{2 - \sqrt{16 - 12}}{0} = \frac{2 - 2}{0} = \frac{0}{0}$ $\newline$ This is an indeterminate form, so we need to use algebraic manipulation to simplify the expression.
Multiply by Conjugate: Multiply the numerator and denominator by the conjugate of the numerator to rationalize the expression. $\newline$ The conjugate of $2$ - \sqrt{ $4$ x - $12$ } is $2$ + \sqrt{ $4$ x - $12$ }. We multiply the numerator and denominator by this conjugate: $\newline$ $\lim_{x \to 4} \frac{(2 - \sqrt{4x - 12})(2 + \sqrt{4x - 12})}{(x - 4)(2 + \sqrt{4x - 12})}$
Apply Difference of Squares: Apply the difference of squares formula to the numerator. $\newline$ $(2 - \sqrt{4x - 12})(2 + \sqrt{4x - 12}) = 2^2 - (\sqrt{4x - 12})^2 = 4 - (4x - 12)$ $\newline$ Simplify the numerator: $\newline$ $4 - 4x + 12 = 16 - 4x$ $\newline$ Now the limit expression is: $\newline$ $\lim_{x \to 4} \frac{16 - 4x}{(x - 4)(2 + \sqrt{4x - 12})}$
Factor Out $-4$ : Factor out $-4$ from the numerator to cancel out the (x - $4$ ) term in the denominator. $\newline$ $\lim_{x \to 4} \frac{-4(x - 4)}{(x - 4)(2 + \sqrt{4x - 12})}$ $\newline$ Now we can cancel out the (x - $4$ ) terms: $\newline$ $\lim_{x \to 4} \frac{-4}{2 + \sqrt{4x - 12}}$
Substitute and Simplify: Substitute x = $4$ into the simplified limit expression. $\newline$ $\lim_{x \to 4} \frac{-4}{2 + \sqrt{4x - 12}} = \frac{-4}{2 + \sqrt{4(4) - 12}} = \frac{-4}{2 + \sqrt{16 - 12}} = \frac{-4}{2 + 2} = \frac{-4}{4} = -1$