evaluate\nlim x→∞ (√(25x² + x) - 5x).\nenter inf for ∞, -inf for -∞, and dne if the limit does not…

Explanation:

Step1: Rationalize the expression

Multiply and divide by $\sqrt{25x^{2}+x}+5x$. [ \begin{align*} &\lim_{x\rightarrow\infty}(\sqrt{25x^{2}+x}-5x)\times\frac{\sqrt{25x^{2}+x}+5x}{\sqrt{25x^{2}+x}+5x}\ =&\lim_{x\rightarrow\infty}\frac{(\sqrt{25x^{2}+x})^{2}-(5x)^{2}}{\sqrt{25x^{2}+x}+5x}\ =&\lim_{x\rightarrow\infty}\frac{25x^{2}+x - 25x^{2}}{\sqrt{25x^{2}+x}+5x}\ =&\lim_{x\rightarrow\infty}\frac{x}{\sqrt{25x^{2}+x}+5x} \end{align*} ]

Step2: Divide numerator and denominator by $x$

Since $x\rightarrow\infty$, $x>0$. So we have: [ \begin{align*} &\lim_{x\rightarrow\infty}\frac{x}{\sqrt{25x^{2}+x}+5x}\ =&\lim_{x\rightarrow\infty}\frac{x/x}{\sqrt{25x^{2}+x}/x + 5x/x}\ =&\lim_{x\rightarrow\infty}\frac{1}{\sqrt{25+\frac{1}{x}}+5} \end{align*} ]

Step3: Evaluate the limit

As $x\rightarrow\infty$, $\frac{1}{x}\rightarrow0$. [ \begin{align*} &\lim_{x\rightarrow\infty}\frac{1}{\sqrt{25+\frac{1}{x}}+5}\ =&\frac{1}{\sqrt{25 + 0}+5}\ =&\frac{1}{5 + 5}\ =&\frac{1}{10} \end{align*} ]

Answer:

$\frac{1}{10}$