Question

Estimating limits numerically and graphically.

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically.

$\underset{x\to \infty }{\lim }\frac{x^5}{e^x}$

Short answer

The value of limit of the given function is$0.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to \infty }{\lim }\frac{x^5}{e^x}$

Step 2. Formula used.

Here, we simplify the fractions.

Step 3. Simplification.

Given,

$\begin{array}{c}\underset{x\to \infty }{\lim }\frac{x^5}{e^x}=\underset{x\to \infty }{\lim }\frac{5x^4}{e^x}\\ =\underset{x\to \infty }{\lim }\frac{5\times 4x^3}{e^x}\\ =\underset{x\to \infty }{\lim }\frac{5\times 4\times 3x^2}{e^x}\\ =\underset{x\to \infty }{\lim }\frac{5\times 4\times 3\times 2\times 1}{e^x}\\ =120\times 0\\ =0\end{array}$

Therefore, our final answer is $0.$