Computer Sciences > GATE 2026 SET-1 > Calculus
Consider the function f: ℝ → ℝ defined as follows:
f(x) = c1ex − c2 loge(1/x), if x > 0
f(x) = 3, otherwise
where c1, c2 ∈ ℝ.
If f is continuous at x = 0, then c1 + c2 = _________. (answer in integer)
f(x) = c1ex − c2 loge(1/x), if x > 0
f(x) = 3, otherwise
where c1, c2 ∈ ℝ.
If f is continuous at x = 0, then c1 + c2 = _________. (answer in integer)
Correct : 3
For f to be continuous at x = 0, the right-hand limit as x → 0+ must equal f(0) = 3.
First simplify the expression for x > 0. Since loge(1/x) = −loge(x), the function becomes f(x) = c1ex + c2loge(x) for x > 0.
Taking the limit as x → 0+:
— c1ex → c1e0 = c1 (finite for any c1)
— c2loge(x) → c2 × (−∞) as x → 0+
For the limit to be finite and equal to 3, the logarithm term must not blow up. The only way to prevent this is to have c2 = 0.
With c2 = 0, the limit becomes simply c1. For continuity this must equal f(0) = 3, so c1 = 3.
Therefore c1 + c2 = 3 + 0 = 3.
Correct answer: 3 ✓
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