Limits, Indeterminate Forms, Continuity & Differentiability
“L'Hôpital's rule, 1^infinity exponential identities, left/right limits, and smoothness conditions.”
1.Formal Limits, Indeterminate Forms, and Smoothness
§ 2.1.Rigorous Definition of Limit: lim_{x -> c} f(x) = L means for every epsilon > 0, there exists delta > 0 such that 0 < |x - c| < delta implies |f(x) - L| < epsilon.
Evaluation Strategies for Indeterminate Forms:
1. Algebraic manipulation / Rationalization: multiply by conjugate.
2. Standard Series Expansions: e^x = 1 + x + x^2/2! + ..., sin x = x - x^3/3! + ..., cos x = 1 - x^2/2! + ...
3. Logarithmic Transformation for Power Forms (1^inf, 0^0, inf^0): Let y = [f(x)]^{g(x)}, then ln y = g(x) * ln f(x). Evaluate lim ln y = L, then lim y = e^L.
Continuity & Differentiability Hierarchy:
- Differentiable at c => Continuous at c.
- Counterexample to converse: f(x) = |x| at x = 0. Continuous (lim = 0 = f(0)), but LHD = -1 != RHD = +1, so not differentiable.
If lim f(x) = 1 and lim g(x) = infinity, then lim [f(x)]^{g(x)} = e^{lim g(x) * [f(x) - 1]}. This single identity solves over 80% of GATE power-limit questions in under 60 seconds.
2. Worked Problem: 1^infinity Indeterminate Form & Continuity Junction
Students often forget to take exp() of the final inner limit when resolving 1^infinity forms.
Problem [Original Practice Problem · Modeled on GATE Pattern]:
Evaluate L = lim_{x -> 0} ( (sin x) / x )^{1 / x^2}.
Step 1: Check indeterminate form:
As x -> 0, (sin x)/x -> 1, and 1/x^2 -> +infinity. Form is 1^infinity.
Step 2: Apply exponential formula L = exp( lim_{x -> 0} [ (sin x)/x - 1 ] * (1/x^2) ):
Inner limit K = lim_{x -> 0} (sin x - x) / x^3.
Step 3: Evaluate K using Taylor expansion:
sin x = x - x^3 / 3! + x^5 / 5! - ... = x - x^3 / 6 + O(x^5)
sin x - x = -x^3 / 6 + O(x^5)
K = lim_{x -> 0} [ -x^3 / 6 ] / x^3 = -1/6.
Step 4: Exponentiate to obtain final limit:
L = e^K = e^{-1/6} = 1 / e^{1/6}.
Primary Academic References & Exam Sources
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