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 limits.
∵ Z =
To get the desired form, we proceed as follows-
Dividing numerator and denominator by tan x-
⇒ Z =
Using algebra of limits-
Z =
Use the formula - (sandwich theorem)
∴ Z =
As, x→ 0
∴ tan x → 0
Let, y = tan x
∴ if x→ 0 ⇒ y→0
Hence, Z can be rewritten as-
Use the formula:
∴ Z = log e = 1
{∵ log e = 1}
Hence,