$$lim_{x
ightarrow0}\frac{sqrt{1 + x}-1}{x}$$

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the numerator
Multiply the fraction by $\frac{\sqrt{1 + x}+1}{\sqrt{1 + x}+1}$.
$$
\begin{align*}
&\lim_{x
ightarrow0}\frac{\sqrt{1 + x}-1}{x}\times\frac{\sqrt{1 + x}+1}{\sqrt{1 + x}+1}\
=&\lim_{x
ightarrow0}\frac{(1 + x)-1}{x(\sqrt{1 + x}+1)}\
=&\lim_{x
ightarrow0}\frac{x}{x(\sqrt{1 + x}+1)}
\end{align*}
$$
## Step2: Simplify the fraction
Cancel out the common factor $x$ in the numerator and denominator.
$$
\begin{align*}
&\lim_{x
ightarrow0}\frac{x}{x(\sqrt{1 + x}+1)}\
=&\lim_{x
ightarrow0}\frac{1}{\sqrt{1 + x}+1}
\end{align*}
$$
## Step3: Substitute $x = 0$
$$
\begin{align*}
&\frac{1}{\sqrt{1+0}+1}\
=&\frac{1}{1 + 1}\
=&\frac{1}{2}
\end{align*}
$$
# Answer:
$\frac{1}{2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the numerator

Step2: Simplify the fraction

Step3: Substitute $x = 0$

Answer: