JEE Maths›Limits, Continuity & Differentiability›Visual Solution

Visual SolutionStandard

Question

\displaystyle\lim_{x \to 0} \frac{e^x - e^{\sin x}}{x - \sin x}

is equal to:

(A)

e

(B)

0

(C)

1

(D)

-1

Solution Path

Factor

e^{\sin x}

from numerator, apply standard limit

\lim (e^t-1)/t = 1

with

t = x - \sin x

. Answer: 1.

01Question Setup

1/4

Evaluate

\displaystyle\lim_{x \to 0} \frac{e^x - e^{\sin x}}{x - \sin x}

. Both numerator and denominator approach 0.

\frac{0}{0}

form

02Factor the Numerator

2/4

Factor out

e^{\sin x}

\frac{e^{\sin x}(e^{x - \sin x} - 1)}{x - \sin x}

. Now we have a standard limit form.

e^{\sin x} \cdot \frac{e^{x-\sin x} - 1}{x - \sin x}

03Standard LimitKEY INSIGHT

3/4

Let

t = x - \sin x \to 0

. Use

\displaystyle\lim_{t \to 0} \frac{e^t - 1}{t} = 1

. Also

e^{\sin x} \to e^0 = 1

. So the limit

= 1 \cdot 1 = 1

\displaystyle\lim \frac{e^t - 1}{t} = 1

04Final Answer

4/4

The limit equals 1. Answer is (C).

\boxed{1}

Concepts from this question2 concepts unlocked

★

L'Hopital's Rule Application

STANDARD

When a limit yields 0/0 or infinity/infinity, differentiate the numerator and denominator separately and re-evaluate the limit

\frac{0}{0} \text{ or } \frac{\infty}{\infty} \implies \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Provides a systematic fallback when standard forms and algebraic tricks fail to resolve indeterminate limits

Indeterminate form limitsExponential limitsLogarithmic limits

Practice (10 Qs) →

Standard Limit Forms

EASY

The seven fundamental limits that serve as building blocks for evaluating all other limits: sin x/x, tan x/x, (1+1/x)^x, (e^x-1)/x, (a^x-1)/x, log(1+x)/x, and (1+x)^(1/x)

\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e

Most JEE limit problems reduce to one of these standard forms after algebraic manipulation or substitution

Direct substitution limitsTrigonometric limitsExponential limits

Practice (14 Qs) →

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