First slide

Introduction to limits

Question

If $L = \lim_{x \to \infty} (x + 1 - \sqrt{{ax}^{2} + x + 3})$ exists finitely the

Moderate

Question

the value of a is

A

-1
B

$\frac{1}{2}$
C

1
D

$\frac{3}{2}$

Solution

$\begin{array}{l} L = \lim_{x \to \infty} (x + 1 - \sqrt{{ax}^{2} + x + 2}) \\ = \lim_{x \to \infty} (\frac{{(x + 1)}^{2} - ({ax}^{2} + x + 3)}{x + 1 + \sqrt{{ax}^{2} + x + 3}}) \\ = \lim_{x \to \infty} (\frac{(1 - a) x^{2} + x - 2}{x + 1 + \sqrt{{ax}^{2} + x = 3}}) \\ L exists finiely if 1 - a = 0 or a = 1 \\ ∴ L = \lim_{x \to \infty} (\frac{x - 2}{x + 1 + \sqrt{x^{2} + x + 3}}) \\ = \lim_{x \to \infty} (\frac{1 - \frac{2}{x}}{1 + \frac{1}{x} + \sqrt{1 + \frac{1}{x} + \frac{3}{x}}}) \\ = \frac{1}{2} \end{array}$

Question

The value of L is

A

$\frac{1}{3}$
B

$\frac{1}{4}$
C

$\frac{1}{2}$
D

$\frac{2}{3}$

Solution

$\begin{array}{l} L = \lim_{x \to \infty} (x + 1 - \sqrt{{ax}^{2} + x + 2}) \\ = \lim_{x \to \infty} (\frac{{(x + 1)}^{2} - ({ax}^{2} + x + 3)}{x + 1 + \sqrt{{ax}^{2} + x + 3}}) \\ = \lim_{x \to \infty} (\frac{(1 - a) x^{2} + x - 2}{x + 1 + \sqrt{{ax}^{2} + x = 3}}) \\ L exists finiely if 1 - a = 0 or a = 1 \\ ∴ L = \lim_{x \to \infty} (\frac{x - 2}{x + 1 + \sqrt{x^{2} + x + 3}}) \\ = \lim_{x \to \infty} (\frac{1 - \frac{2}{x}}{1 + \frac{1}{x} + \sqrt{1 + \frac{1}{x} + \frac{3}{x}}}) \\ = \frac{1}{2} \end{array}$

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