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 y0


Hence, Z can be rewritten as-



Use the formula:


Z = log e = 1


{ log e = 1}


Hence,



38