First slide

Introduction to limits

Question

If $\lim_{n \to \infty} (an - \frac{1 + n^{2}}{1 + n}) = b$ , where a is a finite number, then

Moderate

A

a=1
B

a=0
C

b=1
D

b= -1

Solution

$\begin{array}{l} Limit = \lim_{n \to \infty} \frac{an (1 + n) - (1 + n^{2})}{1 + n} \\ = \lim_{n \to \infty} \frac{(a - 1) n^{2} + an - 1}{n + 1} \\ = \infty if a - 1 \neq 0 \\ If a - 1 = 0, \\ \lim_{n \to \infty} \frac{an - 1}{n + 1} = a = b \\ ∴ a = b = 1 \end{array}$

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