$lim_{x
ightarrow3}\frac{sqrt{x + 1}-2}{x - 3}$

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the numerator
Multiply the fraction by $\frac{\sqrt{x + 1}+2}{\sqrt{x + 1}+2}$.
$$
\begin{align*}
&\lim_{x
ightarrow3}\frac{\sqrt{x + 1}-2}{x - 3}\times\frac{\sqrt{x + 1}+2}{\sqrt{x + 1}+2}\
=&\lim_{x
ightarrow3}\frac{(\sqrt{x + 1})^2-2^2}{(x - 3)(\sqrt{x + 1}+2)}\
=&\lim_{x
ightarrow3}\frac{x + 1-4}{(x - 3)(\sqrt{x + 1}+2)}\
=&\lim_{x
ightarrow3}\frac{x - 3}{(x - 3)(\sqrt{x + 1}+2)}
\end{align*}
$$
## Step2: Simplify the fraction
Cancel out the common factor $(x - 3)$ in the numerator and denominator.
$$
\begin{align*}
&\lim_{x
ightarrow3}\frac{x - 3}{(x - 3)(\sqrt{x + 1}+2)}\
=&\lim_{x
ightarrow3}\frac{1}{\sqrt{x + 1}+2}
\end{align*}
$$
## Step3: Substitute $x = 3$
$$
\begin{align*}
&\frac{1}{\sqrt{3+1}+2}\
=&\frac{1}{\sqrt{4}+2}\
=&\frac{1}{2 + 2}\
=&\frac{1}{4}
\end{align*}
$$
# Answer:
$\frac{1}{4}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the numerator

Step2: Simplify the fraction

Step3: Substitute $x = 3$

Answer: