What is the limit on this equation?

Question

Compute (1-cos(5x))/(x^2)

for values of 0.1, 0.01, and 0.001.

Use these values to guess the exact value of limit as x approaches 0 of (1-cos(5x))/(x^2)

Answer 1

the limit does not exist

Answer 2

f (0.1) = 12.211744

f (0.01) = 12.437396

f (0.001) = 12.439974

The limit as x approaches 0 is 12.5.

Answer 3

the other way to solve is by applying L Hospital rule.

as the it is in the 0/0 form.

differentiate both numerator and denominator

so it will become  5sin5x/2x

again it is in 0/0 form.

again differentiate

=>25cos5x/2

now when x tends to 0.

value becomes 25/2=12.5