Question

find the limit. (limlimits_{x \to 27} \frac{x - 27}{sqrt{x + 9} - 6})

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{x + 9}+6$.

$$\begin{align*} \lim_{x ightarrow27}\frac{x - 27}{\sqrt{x + 9}-6}&=\lim_{x ightarrow27}\frac{(x - 27)(\sqrt{x + 9}+6)}{(\sqrt{x + 9}-6)(\sqrt{x + 9}+6)}\\ \end{align*}$$

Step2: Simplify the denominator

Using the difference of squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the denominator becomes $(\sqrt{x + 9})^{2}-6^{2}=x + 9 - 36=x - 27$.

$$\begin{align*} \lim_{x ightarrow27}\frac{(x - 27)(\sqrt{x + 9}+6)}{x - 27}&=\lim_{x ightarrow27}(\sqrt{x + 9}+6) \quad (x eq27) \end{align*}$$

Step3: Substitute $x = 27$

Substitute $x = 27$ into the expression $\sqrt{x + 9}+6$.

$$ \sqrt{27 + 9}+6=\sqrt{36}+6 = 6 + 6=12 $$

Answer:

12