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 =

To apply the formula of logarithmic limit we need denominator

∴ multiplying in numerator and denominator

Hence, Z can be rewritten as-

Z =

⇒ Z =

{Using algebra of limits}

⇒ Z =

As, x→0 ⇒

Let,

∴ Z =

Use the formula:

∴ Z =

Hence,