Find $\lim _{x \rightarrow 3} \frac{x-3}{\sqrt{4 x+4}-4}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-4$ $\newline$ (B) $1$ $\newline$ (C) $2$ $\newline$ (D) The limit doesn't exist

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Find derivatives of logarithmic functions

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Q. Find

\lim _{x \rightarrow 3} \frac{x-3}{\sqrt{4 x+4}-4}

\newline

Choose

1

answer:

\newline

(A)

-4

\newline

(B)

1

\newline

(C)

2

\newline

(D) The limit doesn't exist

Identify Indeterminate Form: Identify the indeterminate form. $\newline$ We need to evaluate the limit of the function $(x-3)/(\sqrt{4x+4}-4)$ as $x$ approaches $3$ . Let's first plug in the value of $x = 3$ to see if the function is defined at that point or if it results in an indeterminate form. $\newline$ $\lim_{x \to 3}\frac{x-3}{\sqrt{4x+4}-4} = \frac{3-3}{\sqrt{4\cdot 3+4}-4} = \frac{0}{0}$ $\newline$ This is an indeterminate form, so we cannot directly evaluate the limit.
Rationalize Denominator: Rationalize the denominator. $\newline$ To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{4x+4}-4$ is $\sqrt{4x+4}+4$ . $\newline$ $\lim_{x \to 3}\frac{(x-3)(\sqrt{4x+4}+4)}{(\sqrt{4x+4}-4)(\sqrt{4x+4}+4)}$
Apply Difference of Squares: Apply the difference of squares formula. $\newline$ When we multiply the denominator by its conjugate, we get a difference of squares, which simplifies to: $\newline$ $(\sqrt{4x+4}-4)(\sqrt{4x+4}+4) = (\sqrt{4x+4})^2 - (4)^2 = 4x+4 - 16 = 4x - 12$
Simplify Expression: Simplify the expression. $\newline$ Now we simplify the numerator and the denominator separately: $\newline$ Numerator: $(x-3)(\sqrt{4x+4}+4)$ $\newline$ Denominator: $4x - 12$ $\newline$ The limit becomes: $\newline$ $\lim_{x \to 3}\frac{(x-3)(\sqrt{4x+4}+4)}{4x - 12}$
Factor Out Common Term: Factor out the common term in the denominator. $\newline$ We notice that the denominator $4x - 12$ can be factored out as $4(x - 3)$ . This will allow us to cancel out the $(x-3)$ term in the numerator and the denominator. $\newline$ $\lim_{x \to 3}\frac{(x-3)(\sqrt{4x+4}+4)}{4(x - 3)}$
Cancel Common Terms: Cancel out the common terms. $\newline$ After factoring, we can cancel the $(x-3)$ term in the numerator and the denominator: $\newline$ $\lim_{x \to 3}\left(\sqrt{4x+4}+4\right)/4$
Plug in Value: Plug in the value of $x = 3$ . Now that we have simplified the expression and eliminated the indeterminate form, we can safely substitute $x = 3$ into the limit: $\lim_{x \to 3}(\sqrt{4\cdot 3+4}+4)/4 = (\sqrt{12+4}+4)/4 = (\sqrt{16}+4)/4 = (4+4)/4 = 8/4 = 2$