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
∴ we need to take steps to remove this form so that we can get a finite value.
Note: While modifying be careful that you don’t introduce any zero terms in the denominator
As Z =
Multiplying numerator and denominator by √(2+cos x) + 1,we have-
Z =
⇒ Z =
{using a2 – b2 = (a+b)(a-b)}
⇒ Z =
{using basic algebra of limits}
⇒ Z = =
As, 1+cos x = 2cos2(x/2)
∴ Z =
Tip: Similar limit problems involving trigonometric ratios along with algebraic equations are mostly solved using sandwich theorem.
So to solve this problem we need to have a sin term so that we can make use of sandwich theorem.
∵ sin(π/2 – x) = cos x
∴ Z =
As x→π ⇒ π – x → 0
Let y = π – x
Z =
To apply sandwich theorem we have to get the similar form as described below-
∴ Z =
⇒ Z =
Hence,