Q. 54.8( 6 Votes )

Evaluate the following limits:


Answer :

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 xπ 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,



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