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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

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 at the given number a.

g ( t ) = t 2 + 5 t 2 t + 1 , a = 2

To determine

To show: The function g(t)=t2+5t2t+1 is continuous at the number a=2.

Explanation

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

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 definition of continuous, g is continuous at a number 2 if limx2g(t)=g(2).

So proving this, it is enough to show that limx2g(t)=g(2).

Use limit laws to show limx2g(t)=g(2) as follows.

The left hand side of the equality as follows

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