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 =
Adding and subtracting 1 in the numerator to get the desired form
⇒ Z =
⇒ Z =
{using algebra of limits}
To get the desired form to apply the formula we need to divide numerator and denominator by x.
⇒ Z
Using algebra of limits, we have-
Z
Use the formula: and
∴ Z =
{∵ log e = 1}
⇒ Z = 1/2
Hence,