First slide

Introduction to limits

Question

$lim_{x \to \infty} {\sqrt[3]{(x + a) (x + b) (x + c)} - x} =$

Moderate

A

$\sqrt{a b c}$
B

$\frac{a + b + c}{3}$
C

$a b c$
D

${(a b c)}^{\frac{1}{3}}$

Solution

We have,
$\begin{array}{l} lim_{x \to \infty} {\sqrt[3]{(x + a) (x + b) (x + c)} - x} \\ = lim_{x \to \infty} \{\sqrt[3]{(x + a) (x + b) (x + c)} - \sqrt[3]{x^{3}}\} \end{array}$
$= lim_{x \to \infty} \frac{(x + a) (x + b) (x + c) - x^{3}}{{(x + a) (x + b) (x + c)}^{2 / 3} + {\{x^{3} (x + a) (x + b) (x + c)\}}^{1 / 3} + x^{2}}$
$= lim_{x \to \infty} \frac{x^{2} (a + b + c) + x (a b + b c + c a) + a b c}{x^{2} {\{(1 + \frac{a}{x}) (1 + \frac{b}{x}) (1 + c)\}}^{\frac{2}{3}} + x^{2} {\{(1 + \frac{a}{x}) (1 + \begin{matrix} b \\ x \end{matrix}) (1 + \frac{c}{x})\}}^{\frac{1}{3}} + x^{2}}$
$\begin{array}{l} = lim_{x \to \infty} \frac{(a + b + c) + \frac{1}{x} (a b + b c + c a) + \frac{a b c}{x^{2}}}{{\{(1 + \frac{a}{2}) (1 + \frac{b}{x}) (1 + \frac{c}{x})\}}^{3} + \{(1 + \frac{a}{x}) {(1 + (x) (1 + \frac{c}{x})\}}^{\frac{1}{3}} + 1} \\ = \frac{a + b + c}{3} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App