find the following limit. if the limit does n...
find the following limit. if the limit does not exist, click on \does not exist.\ lim(x→3) (4 - √(19 - x))/(x - 3) =
Answer
# Explanation:
## Step1: Rationalize the numerator
Multiply the fraction by $\frac{4 + \sqrt{19 - x}}{4+\sqrt{19 - x}}$.
\[
\begin{align*}
&\lim_{x\rightarrow3}\frac{4-\sqrt{19 - x}}{x - 3}\times\frac{4+\sqrt{19 - x}}{4+\sqrt{19 - x}}\\
=&\lim_{x\rightarrow3}\frac{16-(19 - x)}{(x - 3)(4+\sqrt{19 - x})}\\
=&\lim_{x\rightarrow3}\frac{x - 3}{(x - 3)(4+\sqrt{19 - x})}
\end{align*}
\]
## Step2: Simplify the fraction
Cancel out the common factor $(x - 3)$ in the numerator and denominator.
\[
\begin{align*}
&\lim_{x\rightarrow3}\frac{x - 3}{(x - 3)(4+\sqrt{19 - x})}\\
=&\lim_{x\rightarrow3}\frac{1}{4+\sqrt{19 - x}}
\end{align*}
\]
## Step3: Evaluate the limit
Substitute $x = 3$ into the simplified - function.
\[
\begin{align*}
&\frac{1}{4+\sqrt{19-3}}\\
=&\frac{1}{4+\sqrt{16}}\\
=&\frac{1}{4 + 4}\\
=&\frac{1}{8}
\end{align*}
\]
# Answer:
$\frac{1}{8}$