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Q. 12

Expert-verified
Found in: Page 988

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Fill in the blanks to complete the limit rules. You may assume that $\underset{\mathbf{x}\to \mathbf{a}}{lim} f\left(\mathbf{x}\right)$ and$\underset{\mathbf{x}\to \mathbf{a}}{lim} g\left(\mathbf{x}\right)$ exists and that k is a scalar. $\underset{\mathbf{x}\to \mathbf{a}}{lim} kf\left(\mathbf{x}\right)=$

Ans: $\underset{\mathbf{x}\to \mathbf{a}}{lim} kf\left(\mathbf{x}\right)=k\underset{\mathbf{x}\to \mathbf{a}}{lim} f\left(\mathbf{x}\mathbf{\right)}$ (According to rules of limit.)

See the step by step solution

## Step 1. Given information.

given, $\underset{\mathbf{x}\to \mathbf{a}}{lim} kf\left(\mathbf{x}\right)=$

## Step 2. According to the rules of the limit,

The limit of constant times a function is equal to the constant times the limit of the function.

like if C is a scalar then, $\underset{\mathbf{x}\to \mathbf{a}}{lim} Cf\left(\mathbf{x}\right)=C\underset{\mathbf{x}\to \mathbf{a}}{lim} f\left(\mathbf{x}\mathbf{\right)}$

That's why

$\underset{\mathbf{x}\to \mathbf{a}}{lim} kf\left(\mathbf{x}\right)=k\underset{\mathbf{x}\to \mathbf{a}}{lim} f\left(\mathbf{x}\mathbf{\right)}$