When should I use Taylor Series expansion instead of L'Hôpital's Rule?

Use Taylor expansion whenever the limit involves combinations of functions like $\sin x$, $e^x$, and $\ln(1+x)$ where repeated application of L'Hôpital's Rule creates excessively long terms. Expansions directly expose leading non-zero power terms in a single step.

Why is $\lim_{x \to 0} \frac{\sin x}{x} = 1$ incorrect if $x$ is in degrees?

The derivative $\frac{d}{dx}(\sin x) = \cos x$ assumes radian measure. In degrees, $\sin(x^\circ) = \sin\left(\frac{\pi x}{180}\right)$, which gives $\lim_{x \to 0} \frac{\sin(x^\circ)}{x} = \frac{\pi}{180}$. Always convert angle measures to radians in calculus limits.

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Question #1JEE Main 2024

Evaluate the limit: $\lim_{x \to 0} \left( \frac{\sin x}{x} \right)^{\frac{1}{x^2}}$ .

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Step-by-Step Solution & Concept

Step 1

Identify the limit form. As $x \to 0$ , $\frac{\sin x}{x} \to 1$ and $\frac{1}{x^2} \to \infty$ , which gives the indeterminate form $1^\infty$ .

Step 2

Apply the standard formula $\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]}$ . This transforms the limit into $e^{L}$ , where $L = \lim_{x \to 0} \frac{1}{x^2} \left( \frac{\sin x}{x} - 1 \right) = \lim_{x \to 0} \frac{\sin x - x}{x^3}$ .

Step 3

Use the Taylor expansion $\sin x = x - \frac{x^3}{6} + O(x^5)$ . Substituting yields $L = \lim_{x \to 0} \frac{(x - \frac{x^3}{6}) - x}{x^3} = -\frac{1}{6}$ . Thus, the final value is $e^{-1/6}$ .

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Question #2JEE Main 2023

If $f(x) = \lim_{n \to \infty} \frac{x^{2n} - 1}{x^{2n} + 1}$ , evaluate the right-hand derivative $f'(1^+)$ .

Solution Locked

Step-by-Step Solution & Concept

Step 1

Determine the expression for $f(x)$ for $x > 1$ . Divide numerator and denominator by $x^{2n}$ to get $f(x) = \lim_{n \to \infty} \frac{1 - x^{-2n}}{1 + x^{-2n}}$ .

Step 2

Since $x > 1$ , $x^{-2n} \to 0$ as $n \to \infty$ . Therefore, $f(x) = \frac{1 - 0}{1 + 0} = 1$ for all $x > 1$ .

Step 3

Since $f(x) = 1$ is constant in the right neighborhood of $x = 1$ , its right-hand derivative is $f'(1^+) = \frac{d}{dx}(1) = 0$ .

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Mastering Limits and Derivatives: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Standard Trigonometric Limit

Exponential and Logarithmic Limit

Indeterminate Form $1^\infty$

First Principle of Derivative

L'Hôpital's Rule

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

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