Evaluate the following limits:

As we need to find

We can directly find the limiting value of a function by putting the value of the variable at which the limiting value is asked if it does not take any indeterminate form (0/0 or ∞/∞ or ∞-∞, .. etc.)

Let Z =

∴ We need to take steps to remove this form so that we can get a finite value.

TIP: Most of the problems of logarithmic and exponential limits are solved using the formula and

It also involves a trigonometric term, so there is a possibility of application of Sandwich theorem-

As Z =

As, 1-cos x = 2sin²(x/2)

∴ Z =

⇒ Z =

To get the desired form to apply the formula we need to divide numerator and denominator by x².

⇒ Z =

Using algebra of limits, we have-

Z =

Use the formula: and

∴ Z =

⇒ Z = 2 log 2

Hence,