Mathematics

Mathematics

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. a. (i) Prove that

g(x)

is continuous at

x=0

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

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

. c. (iii) Evaluate

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

. Justify each step of your working.

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