Answer to Question #133449 in Calculus for Kyle Siegfred

Question #133449
lim sin-1 2x/x
x-->0

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

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1
Expert's answer
2020-09-19T13:00:19-0400

For first pat

limx0Sin12xx\lim_{x \to0} \frac{Sin^{-1}2x}{x}


Apply L.hospital rule in above equation,

limx0214x2\lim_{x \to 0} \frac{2}{\sqrt{1-4x^2}}

Now putting the limit

we get

210\frac{2}{\sqrt{1-0}}

=2


Now second part the given limit is

limx0cosecx1x\lim_{x \to0} \frac{cosec x^{-1}}{x}


limx0cosec1xx\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





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