6. find the limit.
$limlimits _{x→∞}(x-sqrt {x^{2}+x})$

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the expression
Multiply and divide by the conjugate $x + \sqrt{x^{2}+x}$
$$
\begin{align*}
\lim_{x
ightarrow\infty}(x - \sqrt{x^{2}+x})&=\lim_{x
ightarrow\infty}\frac{(x - \sqrt{x^{2}+x})(x+\sqrt{x^{2}+x})}{x+\sqrt{x^{2}+x}}\
&=\lim_{x
ightarrow\infty}\frac{x^{2}-(x^{2}+x)}{x+\sqrt{x^{2}+x}}\
&=\lim_{x
ightarrow\infty}\frac{-x}{x+\sqrt{x^{2}+x}}
\end{align*}
$$
## Step2: Divide numerator and denominator by $x$
$$
\begin{align*}
\lim_{x
ightarrow\infty}\frac{-x}{x+\sqrt{x^{2}+x}}&=\lim_{x
ightarrow\infty}\frac{-1}{1+\sqrt{1+\frac{1}{x}}}\
\end{align*}
$$
## Step3: Evaluate the limit
As $x
ightarrow\infty$, $\frac{1}{x}
ightarrow0$
$$
\begin{align*}
\lim_{x
ightarrow\infty}\frac{-1}{1+\sqrt{1+\frac{1}{x}}}&=\frac{-1}{1+\sqrt{1 + 0}}\
&=-\frac{1}{2}
\end{align*}
$$
# Answer:
$-\frac{1}{2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the expression

Step2: Divide numerator and denominator by \(x\)

Step3: Evaluate the limit

Answer: