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→ 0

∴ x-sin x → 0

Let, y = x-sin x

∴ if x→0 ⇒ y→0

Hence, Z can be rewritten as-

Z =

Use the formula:

∴ Z = log e =1

{∵ log e = 1}

Hence,