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

This question is a direct application of limits formula of exponential and logarithmic limits.

To get the desired forms, we need to include mx and nx as follows:

As Z =

⇒ Z =

{using law of exponents}

⇒ Z =

{using algebra of limits}

⇒ Z =

⇒ Z =

To get the form as present in the formula we multiply and divide by 2

∴ Z =

Use the formula:

∴ Z = 2 log a

Hence,