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,