Define the function p(x)= e^ 2x -1 x ,(e^ x -1)

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Define the function

p(x)=\frac{e^{2x}-1}{x\,(e^{x}-1)}, \quad x\neq0.

Answer the following questions. (i) Show that

p(x)

can be simplified for

x\neq0

and write the simplified form. (ii) Examine the limit of

p(x)

x\to0

and hence decide whether the discontinuity at

x=0

is removable or infinite. (iii) Determine all other real numbers, if any, where

p

is not defined. Classify each corresponding discontinuity and find the equations of any vertical asymptotes. (iv) Redefine

p

x=0

, if possible, so that the resulting function is continuous at

x=0

. Justify your answer with appropriate limits.