First slide

Introduction to limits

Question

$If Lt {(1 + \frac{a}{x} + \frac{b}{x^{2}})}^{2 x} = e^{2}, then the values of a and b are$

Moderate

A

$a \in R, b \in R$
B

$a = 1 a n d b = 2$
C

$a \in R, b = 2$
D

$a = 1, b \in R$

Solution

$\begin{array}{l} \lim_{x \to \infty} {(1 + \frac{a}{x} + \frac{b}{x^{2}})}^{2 x} = e \Rightarrow \exp [{Lt}_{x \to \infty} (\frac{a}{x} + \frac{b}{x^{2}}) 2 x] = e^{2} \Rightarrow e^{2 a} = e^{2} \Rightarrow 2 a = 2 \Rightarrow a = 1 \\ ∴ a = 1 and b \in R \end{array}$

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