   Chapter 2.5, Problem 16E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. g ( x ) = x − 1 3 x + 6 ,   ( − ∞ , − 2 )

To determine

To show: The function g(x)=x13x+6 is continuous on the interval (,2).

Explanation

Definitions used:

1. “A function f is continuous at a number a if limxaf(x)=f(a)”.

2. “A function f is continuous on an interval if it is continuous at every number in the interval.

Result used:

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist, then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 4: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 5: limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0

Limit law 6: limxa[f(x)]n=[limxaf(x)]n where n is a positive integer.

Limit law 7: limxac=c

Limit law 8: limxax=a

Limit law 9: limxaxn=an where n is a positive integer.

Limit law 10: limxaxn=an where n is a positive integer [If n is even, assume that a>0].

Limit law 11: limxaf(x)n=limxaf(x)n where n is a positive integer. [If n is even, assume that limxaf(x)>0].

Proof:

By the definition of a continuous function, g is continuous at a number x=a for every a<2 if limxag(x)=g(a)

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