Full solution

Q. Evaluate the limit:

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

Identify Indeterminate Form: Identify the indeterminate form. $\newline$ We need to evaluate the limit of the function as $x$ approaches $12$ . Let's first plug in the value of $x$ into the function to see if it results in an indeterminate form. $\newline$ $\lim_{x \to 12}(\sqrt{x+4}-4)/(x-12) = (\sqrt{12+4}-4)/(12-12) = (\sqrt{16}-4)/0 = (4-4)/0 = 0/0$ $\newline$ This is an indeterminate form, so we cannot directly calculate the limit by substitution.
Algebraic Manipulation: Apply algebraic manipulation to simplify the expression. $\newline$ To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of $\sqrt{x+4}-4$ is $\sqrt{x+4}+4$ . $\newline$ $\lim_{x \to 12}\left(\frac{\sqrt{x+4}-4}{x-12}\right) \cdot \left(\frac{\sqrt{x+4}+4}{\sqrt{x+4}+4}\right) = \lim_{x \to 12}\left(\frac{x+4-16}{(x-12)(\sqrt{x+4}+4)}\right)$
Simplify Expression: Simplify the expression further. $\newline$ Now, we simplify the numerator and the denominator. $\newline$ $\lim_{x \to 12}\left(\frac{x+4-16}{(x-12)(\sqrt{x+4}+4)}\right) = \lim_{x \to 12}\left(\frac{x-12}{(x-12)(\sqrt{x+4}+4)}\right)$ $\newline$ We can see that $(x-12)$ in the numerator and denominator will cancel out, as long as $x \neq 12$ . $\newline$ $\lim_{x \to 12}\left(\frac{1}{\sqrt{x+4}+4}\right)$
Evaluate Limit: Evaluate the limit of the simplified expression. $\newline$ Now that we have simplified the expression, we can substitute $x = 12$ directly into the remaining expression. $\newline$ $\lim_{x \to 12}\left(\frac{1}{\sqrt{x+4}+4}\right) = \frac{1}{\sqrt{12+4}+4} = \frac{1}{\sqrt{16}+4} = \frac{1}{4+4} = \frac{1}{8}$