Let the functions be defined as follows: • • for all real Consider the composition

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let the functions be defined as follows: •

g(x)=\begin{cases}\dfrac{x^{2}-4}{x-2}, & x\neq 2\\3, & x=2\end{cases}

•

f(t)=e^{t}

for all real

t

Consider the composition

F(x)=f\bigl(g(x)\bigr)=e^{g(x)}

. Answer the following, showing all reasoning and citing appropriate mathematical properties where they are used. (a) Calculate

\displaystyle \lim_{x\to 2}g(x)

. Clearly justify each algebraic step in your evaluation. (b) Explain why the function

f

is continuous at the value obtained in part (a). (c) Determine

\displaystyle \lim_{x\to 2}F(x)

, providing a complete justification that references your results from parts (a) and (b).