use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before…

use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim(x→0) (√(19x + 1) - 1)/x select the correct choice below and fill in any answer boxes in your choice. a. lim(x→0) (√(19x + 1) - 1)/x = (type an integer or a simplified fraction.) b. the limit does not exist.

use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim(x→0) (√(19x + 1) - 1)/x select the correct choice below and fill in any answer boxes in your choice. a. lim(x→0) (√(19x + 1) - 1)/x = (type an integer or a simplified fraction.) b. the limit does not exist.

Answer

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{\sqrt{19x + 1}+1}{\sqrt{19x + 1}+1}$. [ \begin{align*} \lim_{x\rightarrow0}\frac{\sqrt{19x + 1}-1}{x}&=\lim_{x\rightarrow0}\frac{(\sqrt{19x + 1}-1)(\sqrt{19x + 1}+1)}{x(\sqrt{19x + 1}+1)}\ \end{align*} ] Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, we get $\lim_{x\rightarrow0}\frac{(19x + 1)-1}{x(\sqrt{19x + 1}+1)}$.

Step2: Simplify the numerator

Simplify the numerator $(19x + 1)-1$ to $19x$. So the limit becomes $\lim_{x\rightarrow0}\frac{19x}{x(\sqrt{19x + 1}+1)}$.

Step3: Cancel out the common factor

Cancel out the common factor $x$ (since $x\neq0$ as we are taking the limit as $x$ approaches 0, not setting $x = 0$). We have $\lim_{x\rightarrow0}\frac{19}{\sqrt{19x + 1}+1}$.

Step4: Apply the limit

Substitute $x = 0$ into the expression $\frac{19}{\sqrt{19x + 1}+1}$. $\frac{19}{\sqrt{19\times0 + 1}+1}=\frac{19}{\sqrt{1}+1}=\frac{19}{2}$.

Answer:

A. $\lim_{x\rightarrow0}\frac{\sqrt{19x + 1}-1}{x}=\frac{19}{2}$