find the limit. (let c and d represent arbitrary real numbers. if the limit is infinite, enter ∞ or -∞, as…

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.)\n lim _ { x \rightarrow infty } left( sqrt { x ^ { 2 } + c x } - sqrt { x ^ { 2 } + d x } \right)

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.)\n lim _ { x \rightarrow infty } left( sqrt { x ^ { 2 } + c x } - sqrt { x ^ { 2 } + d x } \right)

Answer

Explanation:

Step1: Rationalize the expression

Multiply and divide by the conjugate (\sqrt{x^{2}+cx}+\sqrt{x^{2}+dx}): [ \begin{align*} \lim_{x\rightarrow\infty}(\sqrt{x^{2}+cx}-\sqrt{x^{2}+dx})&=\lim_{x\rightarrow\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\rightarrow\infty}\frac{(x^{2}+cx)-(x^{2}+dx)}{\sqrt{x^{2}+cx}+\sqrt{x^{2}+dx}}\ &=\lim_{x\rightarrow\infty}\frac{(c - d)x}{\sqrt{x^{2}+cx}+\sqrt{x^{2}+dx}} \end{align*} ]

Step2: Divide numerator and denominator by (x)

Since (x\rightarrow\infty), (x=\sqrt{x^{2}}). Divide numerator and denominator by (x): [ \begin{align*} \lim_{x\rightarrow\infty}\frac{(c - d)x}{\sqrt{x^{2}+cx}+\sqrt{x^{2}+dx}}&=\lim_{x\rightarrow\infty}\frac{(c - d)}{\sqrt{1+\frac{c}{x}}+\sqrt{1+\frac{d}{x}}} \end{align*} ]

Step3: Evaluate the limit

As (x\rightarrow\infty), (\frac{c}{x}\rightarrow0) and (\frac{d}{x}\rightarrow0). [ \begin{align*} \lim_{x\rightarrow\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})