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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Match List-I with List-II
| List - I | List-II |
| (A) $\lim_{x\to0}(1+2x)^{\frac{1}{x}}$ | (I) $e^{6}$ |
| (B) $\lim_{x\to\infty}(1+\frac{1}{x})^{x}$ | (II) $e^{2}$ |
| (C) $\lim_{x\to0}(1+5x)^{\frac{1}{x}}$ | (III) $e$ |
| (D) $\lim_{x\to\infty}(1+\frac{3}{x})^{2x}$ | (IV) $e^{5}$ |
Solution
Key formulas:
\[
\lim_{x\to 0}(1+ax)^{\frac{1}{x}}=e^{a},\qquad
\lim_{x\to\infty}\left(1+\frac{c}{x}\right)^{kx}=e^{ck}.
\]
Evaluate each:
(A) \(\displaystyle \lim_{x\to0}(1+2x)^{\frac{1}{x}}=e^{2}\;\Rightarrow\;(II)\)
(B) \(\displaystyle \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}=e\;\Rightarrow\;(III)\)
(C) \(\displaystyle \lim_{x\to0}(1+5x)^{\frac{1}{x}}=e^{5}\;\Rightarrow\;(IV)\)
(D) \(\displaystyle \lim_{x\to\infty}\left(1+\frac{3}{x}\right)^{2x}=e^{3\cdot 2}=e^{6}\;\Rightarrow\;(I)\)
✅ Matching: \[ (A)\!\to\!(II),\quad (B)\!\to\!(III),\quad (C)\!\to\!(IV),\quad (D)\!\to\!(I). \]
(A) \(\displaystyle \lim_{x\to0}(1+2x)^{\frac{1}{x}}=e^{2}\;\Rightarrow\;(II)\)
(B) \(\displaystyle \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}=e\;\Rightarrow\;(III)\)
(C) \(\displaystyle \lim_{x\to0}(1+5x)^{\frac{1}{x}}=e^{5}\;\Rightarrow\;(IV)\)
(D) \(\displaystyle \lim_{x\to\infty}\left(1+\frac{3}{x}\right)^{2x}=e^{3\cdot 2}=e^{6}\;\Rightarrow\;(I)\)
✅ Matching: \[ (A)\!\to\!(II),\quad (B)\!\to\!(III),\quad (C)\!\to\!(IV),\quad (D)\!\to\!(I). \]
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