   Chapter 11, Problem 2P

Chapter
Section
Textbook Problem

# A function f is defined by f ( x ) = lim n → ∞ x 2 n − 1 x 2 n + 1 Where is f continuous?

To determine

The points such that the function f(x) is continuous.

Explanation

Given:

The function is f(x)=limnx2n1x2n+1 . (1)

Result used:

If the sequence {rn} ,  then limnrn={  if r>10   if 0<r<1 .

Calculation:

Case (i): |x|<1

If |x|<1 , then 0x2<1 .

Thus, by Result stated above limnx2n=0 .

Substitute 0 for x2n in equation (1),

f(x)=010+1=11=1

Case (ii): |x|=1

That is, x=±1

If x=±1 , then x2=1 .

Therefore, x2n=1 .

Substitute 1 for x2n in equation (1),

f(x)=111+1=02=0

Case (iii): If |x|>1

If |x|>1 , then x2>1

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