Question

The value of $\underset{x\to \infty }{lim}\frac{x^{4}sin\left(\frac{1}{x}\right)+x^3}{1+{\left|x\right|}^3}$ equals

• A. 1
• B. -1
• C. 2
• D. Does not exist

Answer 1

Answer: (A)

1

$\underset{x\to \infty }{lim}\frac{x^{4}sin\left(\frac{1}{x}\right)+x^3}{1+{\left|x\right|}^3}$

$\Rightarrow$ $\underset{x\to \infty }{lim}\frac{x^3\frac{\sin \left(\frac{1}{x}\right)}{\frac{1}{x}}+x^2}{1+|x{|}^2}$

$\Rightarrow$ $\underset{x\to \infty }{lim}\frac{x^3+x^2}{1+|x{|}^3}$

$\Rightarrow$ $\underset{x\to \infty }{lim}\frac{x^2(x+1)}{x^3(\frac{1}{x^3}+1)}$

$\Rightarrow$ $\underset{x\to \infty }{lim}\frac{x+1}{x(\frac{1}{x^3}+1)}$

$\Rightarrow$ $\underset{x\to \infty }{lim}\frac{(1+\frac{1}{x})}{(\frac{1}{x^3}+1)}$

$\Rightarrow$ $\frac{1+\frac{1}{\infty }}{\frac{1}{\infty }+1}$

$\Rightarrow$ $\frac{1}{1}$

$\Rightarrow$ $1$