Solveeit Logo
Login

Question

Question: For x ∈ R, \(\lim_{x \rightarrow 0}\left( \frac{x - 3}{x + 2} \right)^{x}\) =...

For x ∈ R, lim⁡x→0(x−3x+2)x\lim_{x \rightarrow 0}\left( \frac{x - 3}{x + 2} \right)^{x} =

A

e

B

e−1

C

e−5

D

e5

Answer

e−5

Explanation

Solution

For x ∈ R, lim⁡x→∞(x−3x+2)x=lim⁡x→∞{[1−5x+2]x+25}−5xx+2\lim_{x \rightarrow \infty}\left( \frac{x - 3}{x + 2} \right)^{x} = \lim_{x \rightarrow \infty}\left\{ \left\lbrack 1 - \frac{5}{x + 2} \right\rbrack^{\frac{x + 2}{5}} \right\}^{\frac{- 5x}{x + 2}}

= elim⁡x→∞−51+2x=e−5e^{\lim_{x \rightarrow \infty} - \frac{5}{1 + \frac{2}{x} = e^{- 5}}}