Q. 39 - Calculus

Question

Calculate each limit in Exercises 35-80.

$\underset{x\to \infty }{\lim }\left(\sqrt{x}-x\right)$

Step-by-Step Solution

Answer

The limit of the function is $\underset{x\to \infty \hspace{0.17em}}{\lim }\left(\sqrt{x}-x\right)=-\infty \hspace{0.17em}$.

Step 1. Given information.

We have:

$\underset{x\to \infty \hspace{0.17em}}{\lim }\left(\sqrt{x}-x\right)$

Step 2. Find the limits.

The function is $\underset{x\to \infty \hspace{0.17em}}{\lim }\left(\sqrt{x}-x\right).$

Apply the following algebraic property: $a+b=a\left(1+\frac{b}{a}\right)$

$$\begin{gathered} =\underset{x\to \infty \hspace{0.17em}}{\lim }\left(x\left(\frac{1}{\sqrt{x}}-1\right)\right) \\ \underset{x\to a}{\lim }\left[f\left(x\right)·g\left(x\right)\right]=\underset{x\to a}{\lim }f\left(x\right)·\underset{x\to a}{\lim }g\left(x\right) \\ \underset{x\to a}{\lim }\left[f\left(x\right)·g\left(x\right)\right]=\underset{x\to a}{\lim }f\left(x\right)·\underset{x\to a}{\lim }g\left(x\right) \\ =\infty ·\left(-1\right) \\ =-\infty \hspace{0.17em} \end{gathered}$$