find the limit. (let c and d represent arbitrary real numbers. if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)
lim _ { x
ightarrow infty } left( sqrt { x ^ { 2 } + c x } - sqrt { x ^ { 2 } + d x }
ight)

Question

find the limit. (let c and d represent arbitrary real numbers. if the limit is infinite, enter ∞ or -∞, as appropriate. if the limit does not otherwise exist, enter dne.)
 lim _ { x
ightarrow infty } left( sqrt { x ^ { 2 } + c x } - sqrt { x ^ { 2 } + d x }
ight)

Accepted Answer

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

# Answer:
$\frac{c - d}{2}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the expression

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

Step3: Evaluate the limit

Answer: