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,