Limit using L'Hopitale Rule

Question

Find the limit as x approaches zero of 5x - sin (5x) / 5x - tan (5x)

Answer 1

lim(x--->0)[5x-sin(5x)/5x-tan(5x)]

This is in 0/0 form, apply L hospital rule.

= lim(x--->0)[5-5cos(5x)/5-5sec^2(5x)]

= lim(x--->0)[-25sin(5x)/-25tanxsec^2x]

= lim(x--->0)[sin5x/tanxsec^2x]

By taking derivatives from L hospital rule gives

= lim(x--->0) [5cos(5x)/sec^4x+2tan^2xsec^2x]

Evaluting limits.

= 5(1)/(1^4+2*0*1)

= 5