Q. 56 - Calculus

Question

Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.

$\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \left(2-\frac{1}{k}\right)$

Step-by-Step Solution

Answer

$\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \left(2-\frac{1}{k}\right)=2$

Step 1. Given information

$\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \left(2-\frac{1}{k}\right)$

Step 2. Calculate the limit

$\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \left(2-\frac{1}{k}\right)=\left(1+\frac{1}{\infty }\right)\left(2-\frac{1}{\infty }\right)$

$\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \left(2-\frac{1}{k}\right)=1\times 2$

$\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \left(2-\frac{1}{k}\right)=2$