Question

lim sin-1 2x/x
x-->0

lim (csc x- 1/x)
x-->0

show step by step

Answer 1

For first pat

$\lim_{x \to0} \frac{Sin^{-1}2x}{x}$

Apply L.hospital rule in above equation,

$\lim_{x \to 0} \frac{2}{\sqrt{1-4x^2}}$

Now putting the limit

we get

= $\frac{2}{\sqrt{1-0}}$

=2

Now second part the given limit is

$\lim_{x \to0} \frac{cosec x^{-1}}{x}$

$\lim_{x \to0} \frac{cosec \frac{1}{x}}{x}$

This limit does not hold good for L.hospital rule, So the given limit diverges to $\infty$