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Chapter 1: Q. 91 (page 137)

Prove the constant multiple rule for limits:
limx鈫�cf(x)=Landk鈭�鈩�,thenlimx鈫�ckf(x)=kL

Short Answer

Expert verified

Ans:ifx鈭�(c-未,c)鈭�(c,c+未),thenf(x)iswithin蔚|k|ofL

Step by step solution

01

Step 1. Given Information:

limx鈫�cf(x)=Landk鈭�鈩�,thenlimx鈫�ckf(x)=kL

02

Step 2. Prove:

The strategy is to prove the constant multiple rules for limits, considering that

limx鈫�cf(x)=Landk鈭�鈩�,thenlimx鈫�ckf(x)=kL

03

Step 3.Constant multiple rule for limits: 

Giern蔚>0,wecanchoose未>0sothatifx鈭�(c-未,c)鈭�(c,c+未),thenf(x)iswithin蔚kofL.kf(x)-kL=|k(f(x)-L)=|k||f(x)-L|<|k|未=|k|蔚|k|=蔚Hence,iflimx鈫�cf(x)=Landk鈭�鈩�,thenlimx鈫�ckf(x)=kL

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