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,