Mathematics

Problem

Define the function

q(x)=\frac{\tan(4x)}{1+e^{\cos x}}

for all real

x

. (a) Identify the set of real numbers at which

q

is defined. Explain any restrictions on the domain. (b) Justify that

q

is continuous on its entire domain. (c) Use your conclusion from part (b) to evaluate

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

without direct limit computation, and explain the reasoning behind your answer.