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,