If the function

$$f(x)=\left\{\begin{array}{l}\frac{\sin x}{x}+\sec x, x\space\# \space0 \\a\space\ x\ =\ 0\end{array}\right.$$​ is continuous at x = 0, what is the value of a?

Correct option is D

​$$\begin{aligned}&\text{For } f(x) \text{ to be continuous at } x = 0, \text{ the following must hold:} \\&\lim_{x \to 0} f(x) = f(0) \\[10pt]&\text{So, we need to compute:} \\&\lim_{x \to 0} \left( \frac{\sin x}{x} + \sec x \right) \\[15pt]&\textbf{Compute the Limit} \\[5pt]&\text{1. First Term } \left( \frac{\sin x}{x} \right)\text{:} \\&\lim_{x \to 0} \frac{\sin x}{x} = 1 \quad \text{(Standard limit)} \\[10pt]&\text{2. Second Term (} \sec x \text{):} \\&\sec x = \frac{1}{\cos x} \\&\lim_{x \to 0} \sec x = \frac{1}{\cos 0} = 1 \\[10pt]&\text{3. Combined Limit:} \\&\lim_{x \to 0} \left( \frac{\sin x}{x} + \sec x \right) = 1 + 1 = 2 \\[15pt]&\textbf{Assign } a \text{ for Continuity} \\[5pt]&\text{For continuity at } x = 0: \\&f(0) = \lim_{x \to 0} f(x) \Longrightarrow {a = 2}\end{aligned}$$​​