Can anyone help me to evaluate the limit using Hopital's Rule of [ (e^(2x) + e^(-2x) - 2) / 1-cos(pi x) ]?

Question

Can anyone help me to evaluate the limit using Hopital's Rule of [ (e^(2x) + e^(-2x) - 2) / 1-cos(pi x) ] as x approaches 0

please show work

Answer 1

= lim (2e^2x-2e^-2x)/(pi*sin(pix) = lim (4e^2x+4 e^-2x)/pi^2cos(pix)= 8/pi^2

Answer 2

As x ideas 0, the shrink ideas a million^(a million/0). in case you attempt to change x = 0 into this shrink,you get an undefined fraction, a million/0, because of the reality the exponent. you ought to use L'Hopital's Rule right here.

Answer 3

= lim x--> 0 [(2e^(2x) - 2e^(-2x))]/(pisinpix)
= lim x--> 0 [4e^(2x)+4e^(-2x))]/(pi^2 cospix)
= 8/ pi^2