Match List I with List II LIST I LIST II (I) e 3 e 3 e 3 e 3 e 3 (II) 0 (III) 1 (IV) e -1 e - 1 Choose the correct answer from the options given below:

A-IV, B-II, C-III, D-I Explanation: Calculus Limits Solving A. (Matching: IV - ) Standard property ke anusar hota hai. Isliye: B. (Matching: II - ) Numerator aur Denominator ko se divide karne par: C. (Matching: III - ) Ye form hai: D. (Matching: I - ) L'Hopital's Rule ( form) apply karne par: Final Matching Table List-I Solution Value Option Match (A) IV (B) II (C) III (D) I

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CUET PG 2023 Mathematics PYQ — Match List I with List II LIST I LIST II (I) e 3 e 3 e 3 e 3 e 3 … | Mathem Solvex

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CUET PG 2023 Mathematics PYQ — Match List I with List II LIST I LIST II (I) e 3 e 3 e 3 e 3 e 3 … | Mathem Solvex | Mathem Solvex

Explanation

Calculus Limits Solving

A. $\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{\sin x}{x}}$ (Matching: IV - $e^{-1}$ )

Standard property ke anusar $\lim_{x \to 0} \frac{\sin x}{x} = 1$ hota hai. Isliye:

$\text{Limit} = 1^1 = 1$

B. $\lim_{x \to 0} \frac{x + \sin 2x}{x + 2x^2}$ (Matching: II - $0$ )

Numerator aur Denominator ko $x$ se divide karne par:

$\lim_{x \to 0} \frac{1 + \frac{\sin 2x}{x}}{1 + 2x}$

$\text{Limit} = \frac{1 + 2 \left(\frac{\sin 2x}{2x}\right)}{1 + 2(0)} = \frac{1+2}{1} = 3$

C. $\lim_{x \to 0} (e^{2x} + x)^{\frac{1}{x}}$ (Matching: III - $1$ )

Ye $1^\infty$ form hai:

$\text{Limit} = e^{\lim_{x \to 0} \frac{1}{x}(e^{2x} + x - 1)}$

$\text{Limit} = e^{\lim_{x \to 0} \left(\frac{e^{2x}-1}{x} + \frac{x}{x}\right)}$

$\text{Limit} = e^{2+1} = e^3$

D. $\lim_{x \to a} \frac{\log(x - a)}{\log(e^x - e^a)}$ (Matching: I - $e^3$ )

L'Hopital's Rule ( $0/0$ form) apply karne par:

$\lim_{x \to a} \frac{\frac{d}{dx}\log(x-a)}{\frac{d}{dx}\log(e^x - e^a)} = \lim_{x \to a} \frac{\frac{1}{x-a}}{\frac{1}{e^x - e^a} \cdot e^x}$

$\text{Limit} = \lim_{x \to a} \frac{e^x - e^a}{e^x(x - a)} = \lim_{x \to a} \frac{e^x}{e^x + e^x(x-a)} = \frac{e^a}{e^a} = 1$

Final Matching Table

List-I Solution Value Option Match

(A) $1$ IV

(B) $3$ II

(C) $e^3$ III

(D) $1$ I

LIST I	LIST II
$(A) Value of lim_{x \to 0} (\frac{sin x}{x})^{\frac{s i n x}{x}}$	(I) e³
$(B) Value of lim_{x \to 0} \frac{x + sin 2 x}{x + 2 x ^{2}}$	(II) 0
$(C) Value of lim_{x \to 0} (e^{2 x} + x)^{\frac{1}{x}}$	(III) 1
$(D) Value of lim_{θ \to 0} \frac{lo g ( x - a )}{lo g ( e ^{x} - a )} of$	(IV) e^-1

CUET PG 2023 — Mathematics PYQ

Choose the correct answer:

Explanation

Calculus Limits Solving

Final Matching Table

Explanation

Calculus Limits Solving

Final Matching Table

List-I	Solution Value	Option Match
(A)	$1$	IV
(B)	$3$	II
(C)	$e^3$	III
(D)	$1$	I