Question:
\[
\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=
\]
\[
\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=
\]
Show Hint
Always remember the standard limit \(\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e\). It is used frequently in limits.
Updated On: May 7, 2026
- \(0\)
- \(1\)
- \(e\)
- \(\infty\)
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The Correct Option is C
Solution and Explanation
Concept:
A very important standard limit is:
\[
\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e
\]
Step 1: Compare the given expression with the standard limit. \[ \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x \] This is exactly the standard exponential limit.
Step 2: Therefore, directly using the standard result: \[ \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e \] Hence, the required value is: \[ \boxed{e} \]
Step 1: Compare the given expression with the standard limit. \[ \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x \] This is exactly the standard exponential limit.
Step 2: Therefore, directly using the standard result: \[ \lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e \] Hence, the required value is: \[ \boxed{e} \]
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