First slide

Introduction to limits

Question

$If lim_{x \to \infty} {(1 + \frac{λ}{x} + \frac{μ}{x^{2}})}^{2 x} = e^{2} then$

Moderate

A

$λ = 1, μ = 2$
B

$λ = 2, μ = 1$
C

$λ = 1, μ = any real constant$
D

$λ = μ = 1$

Solution

Given limit is
$\begin{array}{l} \Rightarrow e^{lim_{x \to \infty} [1 + \frac{λ}{x} + \frac{μ}{x^{2}} - 1] 2 x} = e^{2} \\ \Rightarrow e^{2 lim_{x \to \infty} [λ + \frac{μ}{x}]} = e^{2} \\ \Rightarrow e^{2 λ} = e^{2} \\ \Rightarrow λ = 1, μ \in R \end{array}$

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