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
As Z
To apply the formula of logarithmic limits we need to get the form that matches with one in formula
∴ We proceed as follows-
Z
⇒ Z
⇒ Z
∵ x→a ⇒ x/a →1
⇒ x/a – 1 → 0
Let, (x/a)-1 = y
∴ y→0
Hence, Z can be rewritten as-
Z
Use the formula:
∴ Z
Hence,