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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

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

If neither lim x a f ( x ) nor lim x a g ( x ) exists , then lim x a [ f ( x ) + g ( x ) ] does not exist.

To determine

Whether the statement, “if neither limxaf(x) nor limxag(x) exists, then limxa[f(x)+g(x)] does not exist” is true or false.

Explanation

The given statement is false since the below example disproves the given statement.

Let f(x)=1x2  and g(x)=1x2 and a=0.

Compute the limit of the functions as x approaches 0. That is, limx0(1x2) and limx0(1x2).

As x sufficiently close to 0 but x0, then the values of the functions f(x) and g(x) can be arbitrarily large.

Thus, the limit value of the functions does not exist.

Consider limxa[f(x)+g(x)]

limxa[f(x)+g(x)

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