Q. 1

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 Z


we need to take steps to remove this form so that we can get a finite value.


Tip: Similar limit problems involving trigonometric ratios 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.


Note: While modifying be careful that you don’t introduce any zero terms in the denominator


As,


Multiplying numerator and denominator by 1-cos x, We have-



Z =


{As 1-cos2x = sin2x}


Z =





To apply sandwich theorem, we need to have limit such that variable tends to 0 and following forms should be there


Here x π so we need to do modifications before applying the theorem.


As, sin (π-x) = sin x or sin (x - π) = -sin x and tan(π – x) = -tan x


we can say that-


sin2x = sin2(x-π) and tan2x = tan2(x-π)


As x π


(x – π) 0


Let us represent x - π with y


Z =


Dividing both numerator and denominator by y2


Z =


Z = {Using basic limits algebra}


As,


Z =



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