GATE CS 2026 Set 1 Q.32: Calculus

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows:

f(x) = \begin{cases} c_1 e^x - c_2 \log_e\left(\frac{1}{x}\right), & \text{if } x > 0 \\ 3, & \text{otherwise} \end{cases}

where $c_1, c_2 \in \mathbb{R}$ .

If $f$ is continuous at $x = 0$ , then $c_1 + c_2 =$ _________. (answer in integer)

Your answer

Official answer and solution

Official answer: 3
From: Official GATE 2026 answer key

Solution

Equate the right-hand limit $\lim_{x \to 0^+} f(x)$ with $f(0) = 3$ .

For $f$ to be continuous at $x = 0$ , the right-hand limit must exist and equal $f(0) = 3$ :

\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \left(c_1 e^x + c_2 \ln x\right) = 3

Since $\lim_{x \to 0^+} e^x = 1$ and $\lim_{x \to 0^+} \ln x = -\infty$ , the limit is finite if and only if $c_2 = 0$ . Substituting $c_2 = 0$ gives:

\lim_{x \to 0^+} c_1 e^x = c_1 = 3

Therefore, $c_1 + c_2 = 3 + 0 = 3$ .

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 32 of 65
Section: Computer Science and Information Technology

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