Limits and Discontinuity

Question

(a) What type of discontinuity does the function 𝑔(𝑥)=1/𝑒^1/x -1have at 𝑥=0

(b) For which value of 𝑐 will the function 𝑓(𝑥)=(1/𝑒^1/x−1+𝑐)² have a limit at 𝑥=0?

(Hint: Use one-sided limits.)

(c) Using the value found in part (b), how would you define 𝑓(𝑥)at 𝑥=0 so that 𝑓 becomes continuous at 𝑥=0?

Answer 1

(a) What type of discontinuity does the function 𝑔(𝑥) = 1/(𝑒^1/x – 1) have at 𝑥 = 0

As x→0, the left-hand side limit (-1) and right-hand side limit (0) are different. So it's a jump discontinuity.

(b) For which value of 𝑐 will the function 𝑓(𝑥) = (1/(𝑒^1/x − 1) + 𝑐)² have a limit at 𝑥=0?

For c = 0 there is a jump discontinuity

where the left-hand side limit = 0 and where the right-hand side limit = 1

When c = 1/2, the left-hand side and right-hand side limits both equal 1/4

So 𝑓(𝑥) has a limit when c = 1/2

(c) Using the value found in part (b), how would you define 𝑓(𝑥) at 𝑥 = 0 so that 𝑓 becomes continuous at 𝑥=0?

You define 𝑓(𝑥) = 1/4 at 𝑥 = 0