evaluate
lim(x→∞)(√(4x² + x) - 2x).
enter inf for ∞, -inf for -∞, and dne if the limit does not exist.
limit =

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the expression
Multiply and divide by the conjugate $\sqrt{4x^{2}+x}+2x$.
$$
\begin{align*}
&\lim_{x
ightarrow\infty}(\sqrt{4x^{2}+x}-2x)\times\frac{\sqrt{4x^{2}+x}+2x}{\sqrt{4x^{2}+x}+2x}\
=&\lim_{x
ightarrow\infty}\frac{(\sqrt{4x^{2}+x})^{2}-(2x)^{2}}{\sqrt{4x^{2}+x}+2x}\
=&\lim_{x
ightarrow\infty}\frac{4x^{2}+x - 4x^{2}}{\sqrt{4x^{2}+x}+2x}\
=&\lim_{x
ightarrow\infty}\frac{x}{\sqrt{4x^{2}+x}+2x}
\end{align*}
$$
## Step2: Divide numerator and denominator by $x$
Since $x
ightarrow\infty$, $x>0$. So we have:
$$
\begin{align*}
&\lim_{x
ightarrow\infty}\frac{x}{\sqrt{4x^{2}+x}+2x}\
=&\lim_{x
ightarrow\infty}\frac{x/x}{\sqrt{4x^{2}+x}/x + 2x/x}\
=&\lim_{x
ightarrow\infty}\frac{1}{\sqrt{\frac{4x^{2}+x}{x^{2}}}+2}\
=&\lim_{x
ightarrow\infty}\frac{1}{\sqrt{4+\frac{1}{x}}+2}
\end{align*}
$$
## Step3: Evaluate the limit
As $x
ightarrow\infty$, $\frac{1}{x}
ightarrow0$.
$$
\begin{align*}
&\lim_{x
ightarrow\infty}\frac{1}{\sqrt{4+\frac{1}{x}}+2}\
=&\frac{1}{\sqrt{4 + 0}+2}\
=&\frac{1}{2 + 2}\
=&\frac{1}{4}
\end{align*}
$$
# Answer:
$\frac{1}{4}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the expression

Step2: Divide numerator and denominator by $x$

Step3: Evaluate the limit

Answer: