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.
As Z =
⇒ Z =
⇒ Z =
{using properties of exponents}
⇒ Z =
{using algebra of limits}
⇒ Z =
∴ Z =
As, x→ (π/2)
∴ cot(π/2) – cos(π/2) → 0
Let, y = cot x – cos x
∴ if x→π/2 ⇒ y→0
Hence, Z can be rewritten as-
Z =
Use the formula:
∴ Z = log a
Hence,