Consider the function f R → R defined as follows: f(x) = c1 ex - c2 loge ((1)/(x) ), & if x > 0 3 & otherwise where c1, c2 ∈ R. If f is continuous at x = 0, then c1 + c2 = … (answer in integer)

Official answer: 3. Explanation: For f(x) to be continuous at x = 0, the right-hand limit must equal the value of the function at x = 0: limx → 0+ f(x) = f(0) = 3 For x > 0, we have: f(x) = c1 ex - c2 loge ((1)/(x) ) = c1 ex + c2 loge(x) As x → 0+: - ex → e0 = 1 - loge(x) → -∞ For limx → 0+ f(x) to exist and be finite, the unbounded term must vanish, which requires: c2 = 0 With c2 = 0: limx → 0+ f(x) = limx → 0+ c1 ex = c1(1) = c1 Equating to f(0) = 3: c1 = 3 Therefore: c1 + c2 = 3 + 0 = 3

Calculus

GATE CSE 2026 Set 1 · Q32

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Consider the function $f\colon \mathbb{R} \to \mathbb{R}$ 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)

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Official answer: 3

Calculus

For $f(x)$ to be continuous at $x = 0$ , the right-hand limit must equal the value of the function at $x = 0$ :

\lim_{x \to 0^+} f(x) = f(0) = 3

For $x > 0$ , we have:

f(x) = c_1 e^x - c_2 \log_e\left(\frac{1}{x}\right) = c_1 e^x + c_2 \log_e(x)

As $x \to 0^+$ :

$e^x \to e^0 = 1$
$\log_e(x) \to -\infty$

For $\lim_{x \to 0^+} f(x)$ to exist and be finite, the unbounded term must vanish, which requires:

c_2 = 0

With $c_2 = 0$ :

\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} c_1 e^x = c_1(1) = c_1

Equating to $f(0) = 3$:

c_1 = 3

Therefore:

c_1 + c_2 = 3 + 0 = 3

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This Calculus previous year question is Q32 of the GATE CSE 2026 Set 1 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

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Calculus in GATE CSE

Calculus has 40 GATE CSE PYQs across 25 papers from 2011 to 2026. 11 of them are 1-mark NAT questions like this one, and GATE CSE 2026 Set 1 has 2.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.