lim_(x rarr-2)(sqrt(3x+10)-2)/(x+2)=

Identify Indeterminate Form: Identify the indeterminate form. Substitute x with -2 in the function to see if it results in an indeterminate form. (sqrt(3(-2)+10)-2)/((-2)+2) = (sqrt(6)-2)/0 This is an indeterminate form of type 0/0.Rationalize the Numerator: Rationalize the numerator. To resolve the indeterminate form, we can rationalize the numerator by multiplying the numerator and the denominator by the conjugate of the numerator. Conjugate of sqrt(3x+10)-2 is sqrt(3x+10)+2. (sqrt(3x+10)-2)/(x+2) * (sqrt(3x+10)+2)/(sqrt(3x+10)+2)Apply Conjugate: Apply the conjugate. Multiply the numerators and the denominators. ((sqrt(3x+10)-2)(sqrt(3x+10)+2))/((x+2)(sqrt(3x+10)+2)) = ((3x+10) - (2^2))/((x+2)(sqrt(3x+10)+2)) = (3x+10 - 4)/((x+2)(sqrt(3x+10)+2)) = (3x+6)/((x+2)(sqrt(3x+10)+2))Simplify the Expression: Simplify the expression. Notice that (3x+6) can be factored as 3(x+2). 3(x+2)/((x+2)(sqrt(3x+10)+2)) Now, we can cancel out the (x+2) terms in the numerator and the denominator. 3/((sqrt(3x+10)+2))Evaluate the Limit: Evaluate the limit. Now that the indeterminate form is resolved, substitute x with -2. 3/((sqrt(3(-2)+10)+2)) = 3/((sqrt(6)+2))Calculate Final Value: Calculate the final value. 3/((sqrt(6)+2)) This is the final value of the limit as x approaches -2.

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Grade 7

Identify equivalent linear expressions I

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\lim _{x \rightarrow-2} \frac{\sqrt{3 x+10}-2}{x+2}=

Identify Indeterminate Form: Identify the indeterminate form. $\newline$ Substitute $x$ with $-2$ in the function to see if it results in an indeterminate form. $\newline$ $\frac{\sqrt{3(-2)+10}-2}{(-2)+2}$ $\newline$ = $\frac{\sqrt{6}-2}{0}$ $\newline$ This is an indeterminate form of type $\frac{0}{0}$ .
Rationalize the Numerator: Rationalize the numerator. $\newline$ To resolve the indeterminate form, we can rationalize the numerator by multiplying the numerator and the denominator by the conjugate of the numerator. $\newline$ Conjugate of $\sqrt{3x+10}-2$ is $\sqrt{3x+10}+2$ . $\newline$ $\frac{\sqrt{3x+10}-2}{x+2} \cdot \frac{\sqrt{3x+10}+2}{\sqrt{3x+10}+2}$
Apply Conjugate: Apply the conjugate. $\newline$ Multiply the numerators and the denominators. $\newline$ $\frac{(\sqrt{3x+10}-2)(\sqrt{3x+10}+2)}{(x+2)(\sqrt{3x+10}+2)}$ $\newline$ = $\frac{(3x+10) - (2^2)}{(x+2)(\sqrt{3x+10}+2)}$ $\newline$ = $\frac{3x+10 - 4}{(x+2)(\sqrt{3x+10}+2)}$ $\newline$ = $\frac{3x+6}{(x+2)(\sqrt{3x+10}+2)}$
Simplify the Expression: Simplify the expression. $\newline$ Notice that $(3x+6)$ can be factored as $3(x+2)$ . $\newline$ $\frac{3(x+2)}{(x+2)(\sqrt{3x+10}+2)}$ $\newline$ Now, we can cancel out the $(x+2)$ terms in the numerator and the denominator. $\newline$ $\frac{3}{(\sqrt{3x+10}+2)}$
Evaluate the Limit: Evaluate the limit. $\newline$ Now that the indeterminate form is resolved, substitute $x$ with $-2$ . $\newline$ $\frac{3}{(\sqrt{3(-2)+10}+2)}$ $\newline$ = $\frac{3}{(\sqrt{6}+2)}$
Calculate Final Value: Calculate the final value. $\frac{3}{(\sqrt{6}+2)}$ This is the final value of the limit as $x$ approaches $-2$ .