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 algebra of limits}
⇒ Z =
⇒ Z = {∵ sin(x-π/2) = -cos x}
As x→π/2
∴ x-π/2→0
Let x-π/2 = y and y→0
Z can be rewritten as-
Z =
Dividing numerator and denominator by sin y to get the form present in the formula
Z =
Using algebra of limits:
Z =
Use the formula: and
∴ Z =
Hence,