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
To get rid of indeterminate form we will divide numerator and denominator by sin x
∴ Z
Using Algebra of limits we have-
Z
Where, A
and B
{from sandwich theorem}
As A
Let, sin x =y
As x→0 ⇒ y→0
∴ A
Using
A = log e = 1
∴
Hence,