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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

If

f ( x ) = { x 2 if x is rational 0 if x is irrational

prove that lim x 0 f ( x ) = 0.

To determine

To prove: The value of limx0f(x)=0 where f(x)={x2if x is rational0if x is irrational.

Explanation

Theorem used:

The Squeeze Theorem

“If f(x)g(x)h(x) when x is near a (except possibly at a) and limxaf(x)=limxah(x)=L then limxag(x)=L.”

Limit Laws:

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

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

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

Proof:

Let f(x)={x2if x is rational0if x is irrational.

From this piecewise function, it is clear that 0f(x)x2.

Apply the Squeeze Theorem on the function f(x).

When x approaches to zero, the inequality becomes, limx00limx0f(x)limx0x2

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