Understanding the limit of trig functions:
We already know that limit is defined by the value a function attains when the index number approaches a particular point. In the similar fashion, limits of trigonometric function can be known by the value they attain when its input index reaches the given value. The limits of trig functions can also be obtained from the graphs of this functions.
The examples given below will help to understand the limit of trig functions.
(1) Find the limit for
Here we need to calculate the limit when index x approaches 0. Hence putting x = 0 in above trigonometric function, we know that cos 0 = 1 and sin 0 = 0. Thus limit L = 1 / (0 – 3) = -1/3. Thus the limit of above function for x -> 0 is -1/3.
Since x -> 0+, value of cot x has to be in first quadrant i.e. positive values. Now cot x = cos x / sin x. Hence cot 0 = cos 0 / sin 0 = 1 / 0 = ∞. Hence there is need to compute the denominator. The limit of this function is a positive infinity. Thus this function will be having a vertical asymptote at x = 0.
If we directly find the limit of above function, it comes to 0 / 0 = ∞. But we have a formula for limit which says that limit of a function f of x divided by x is 1, i.e .Here f(x) = sin x. Thus using this formula, we get the limit of this function as 1.
Thus we have understood the theory for finding limits of trigonometric functions and solved some examples related to it. This limit properties can be used as a basis for derivatives, integration and for other applications.
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