Find the limits of the follwing...or if relevant, show that they do not exist:-?

Question

i) lim (coshx-1)/x^2

x ->0

ii) lim (cosh^-1 x) / (x-1)^1/2

x->1+

Answer 1

Since these are undefined quotients at the required value of x then use L'Hopital's rule. This says differentiate top and bottom separately until you get a defined limit and that is the limit of the original quotient.

1) (coshx - 1)/(x^2) becomes sinhx/2x becomes coshx/2

This has value 1/2 when x = 0 so that is limiting value of original function.

I'll leave you to do the same sort of thing with the other one.