find the limit.\n\\( \\lim _{x \\rightarrow 225} \\frac{\\sqrt{x}-15}{x - 225} \\)\nselect the correct…

find the limit.\n\\( \\lim _{x \\rightarrow 225} \\frac{\\sqrt{x}-15}{x - 225} \\)\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. \\( \\lim _{x \\rightarrow 225} \\frac{\\sqrt{x}-15}{x - 225}= \\) (type an integer or a simplified fraction.)\nb. the limit does not exist.

find the limit.\n\\( \\lim _{x \\rightarrow 225} \\frac{\\sqrt{x}-15}{x - 225} \\)\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. \\( \\lim _{x \\rightarrow 225} \\frac{\\sqrt{x}-15}{x - 225}= \\) (type an integer or a simplified fraction.)\nb. the limit does not exist.

Answer

Explanation:

Step1: Rationalize the numerator

Multiply the numerator and denominator by $\sqrt{x}+15$. $$ \begin{align*} \lim_{x\rightarrow225}\frac{\sqrt{x}-15}{x - 225}&=\lim_{x\rightarrow225}\frac{(\sqrt{x}-15)(\sqrt{x}+15)}{(x - 225)(\sqrt{x}+15)}\ \end{align*} $$ Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $x-225$. So we have $\lim_{x\rightarrow225}\frac{x - 225}{(x - 225)(\sqrt{x}+15)}$

Step2: Simplify the expression

Cancel out the common factor $(x - 225)$ (for $x\neq225$, as $x\rightarrow225$ but $x\neq225$ in the limit process). We get $\lim_{x\rightarrow225}\frac{1}{\sqrt{x}+15}$

Step3: Substitute $x = 225$

Substitute $x = 225$ into $\frac{1}{\sqrt{x}+15}$. When $x = 225$, $\sqrt{x}=\sqrt{225}=15$. Then $\frac{1}{\sqrt{225}+15}=\frac{1}{15 + 15}=\frac{1}{30}$

Answer:

A. $\lim_{x\rightarrow225}\frac{\sqrt{x}-15}{x - 225}=\frac{1}{30}$