Question

The value of lim        ((tanx-x)÷x)×sin(1/x²)  
                    x->0
a) does not exist    
b)equal to 1
c)equal to zero
d) exist but is diffrent from 0 and 1

Answer 1

c) equal to zero.

Let y = 1/x,  as x -> 0 ,  y-> infinity.

$\lim_{x \to 0} ( \frac{tanx-x}{x}) * \lim_{y \to \infty} sin(y^2)\\ \\ \lim_{x \to 0} ( \frac{tanx-x}{x})= \lim_{x \to 0} \frac{tanx}{x}- \lim_{x \to 0} \frac{x}{x}\\ \\= \lim_{x \to 0} \frac{Sin\ x}{x}*Cos\ x- \lim_{x \to 0} 1 = 1 *Cos\ 0 - 1 = 0\\ \\\lim_{y \to \infty} sin(y^2) = not\ convergent\ to\ a\ value,\ always\ between\ 0\ and\ 1.$

As product of zero and another real number (between -1 and 1) will be zero always, the answer is zero.