Consider the function defined by

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

Problem

Consider the function

g:\mathbb{R}\to\mathbb{R}

defined by

g(x)=e^{x^{2}}\ln(x+1)+4x^{3}

. Answer the following. (i) Prove that

g(x)

is continuous at

x=0

. (ii) Using the result of part (i), find

\displaystyle \lim_{x \to 0} g(x)

. (iii) Evaluate

\displaystyle \lim_{x \to 0}\bigl(5g(x)-g(0)\bigr)

. Justify each step of your working.