Answer the following questions related to limits of exponential and logarithmic functions.

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

Problem

Answer the following questions related to limits of exponential and logarithmic functions. (i) Explain why

\displaystyle \lim_{x\to 1} a^{x}=a

for every real

a>0,\;a\neq 1

. Your explanation should refer to an appropriate property of limits. (ii) Using the result of part (i) and standard limit laws, justify that

\lim_{x\to 1} (x^{2}-1)\,\log_{a}x=0,

where

a>0,\;a\neq 1

. Provide full reasoning for each step.