Answer
$$\lim\limits_{x\to\infty}(\sqrt{x^2+4x+1}-x)=2$$
Work Step by Step
$$A=\lim\limits_{x\to\infty}(\sqrt{x^2+4x+1}-x)$$
Multiply the function by $\frac{\sqrt{x^2+4x+1}+x}{\sqrt{x^2+4x+1}+x}$, we have:
$$A=\lim\limits_{x\to\infty}\frac{(\sqrt{x^2+4x+1}-x)(\sqrt{x^2+4x+1}+x)}{\sqrt{x^2+4x+1}+x}$$
- Consider the numerator:
$(\sqrt{x^2+4x+1}-x)(\sqrt{x^2+4x+1}+x)=(x^2+4x+1)-x^2=4x+1$
So, $$A=\lim\limits_{x\to\infty}\frac{4x+1}{\sqrt{x^2+4x+1}+x}$$
Divide both numerator and denominator by $x$, we have
- Numerator: $$\frac{4x+1}{x}=4+\frac{1}{x}$$
- Denominator:
As $x\to\infty$, we only care for the values of $x\gt0$, which means $\sqrt{x^2}=|x|=x$
So, $$\frac{\sqrt{x^2+4x+1}+x}{x}=\frac{\sqrt{x^2+4x+1}}{x}+1=\sqrt{\frac{x^2+4x+1}{x^2}}+1=\sqrt{1+\frac{4}{x}+\frac{1}{x^2}}+1$$
Therefore, $$A=\lim\limits_{x\to\infty}\frac{4+\frac{1}{x}}{\sqrt{1+\frac{4}{x}+\frac{1}{x^2}}+1}$$$$A=\frac{4+0}{\sqrt{1+4\times0+0}+1}$$$$A=2$$
