evaluate\nlim(x→∞)(√(4x² + x) - 2x).\nenter inf for ∞, -inf for -∞, and dne if the limit does not…

Explanation:

Step1: Rationalize the expression

Multiply and divide by the conjugate $\sqrt{4x^{2}+x}+2x$. [ \begin{align*} &\lim_{x\rightarrow\infty}(\sqrt{4x^{2}+x}-2x)\times\frac{\sqrt{4x^{2}+x}+2x}{\sqrt{4x^{2}+x}+2x}\ =&\lim_{x\rightarrow\infty}\frac{(\sqrt{4x^{2}+x})^{2}-(2x)^{2}}{\sqrt{4x^{2}+x}+2x}\ =&\lim_{x\rightarrow\infty}\frac{4x^{2}+x - 4x^{2}}{\sqrt{4x^{2}+x}+2x}\ =&\lim_{x\rightarrow\infty}\frac{x}{\sqrt{4x^{2}+x}+2x} \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{4x^{2}+x}+2x}\ =&\lim_{x\rightarrow\infty}\frac{x/x}{\sqrt{4x^{2}+x}/x + 2x/x}\ =&\lim_{x\rightarrow\infty}\frac{1}{\sqrt{\frac{4x^{2}+x}{x^{2}}}+2}\ =&\lim_{x\rightarrow\infty}\frac{1}{\sqrt{4+\frac{1}{x}}+2} \end{align*} ]

Step3: Evaluate the limit

As $x\rightarrow\infty$, $\frac{1}{x}\rightarrow0$. [ \begin{align*} &\lim_{x\rightarrow\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:

$\frac{1}{4}$