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

∴ Z

{using (a-b)² = a²+b²-2ab}

Z

To apply the formula we need to bring the exact form present in the formula, so-

Z

{Adding and subtracting 1 in numerator}

⇒ Z

Using algebra of limits-

Z

Use the formula:

∴ Z =

Hence,