Evaluate the limit as x approaches infinity of ((x + 2) / (x + 1))^(2x + 3).

B. e^2 — This is of the form 1^infinity. We use the formula limit (1 + 1/f(x))^g(x) = e^limit(g(x)/f(x)). Here, the expression is (1 + 1/(x+1))^(2x+3), which leads to e^limit((2x+3)/(x+1)) = e^2.

JEE Main · MathematicsMedium

$lim_{x \to \infty} (\frac{x + 2}{x + 1})^{2 x + 3}$

A. $e$
B. $e^{2}$
C. $e^{3}$
D. $1$

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Correct answer: B — $e^{2}$

Solution

x \to \infty lim (1 + \frac{1}{x + 1})^{2 x + 3} = e^{l i m_{x \to \infty} \frac{2 x + 3}{x + 1}} = e^{2} .

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limx→∞​(x+1x+2​)2x+3

Solution

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$lim_{x \to \infty} (\frac{x + 2}{x + 1})^{2 x + 3}$