Solved Examples on Limits of Trigonometric Functions

Example 1: Given: g(x) = cos(x)/x, Find: lim_x→∞ g(x)

Solution: 

We know,

-1≤ cos(x)≤ 1 under its domain    

Dividing by x,

-1/x ≤ cos(x)/x ≤ 1/x

Then let f(x) = -1/x and h(x) = 1/x          

f(x) ≤ g(x) ≤ h(x)              

Using sandwich theorem,

As lim_x→a h(x) = lim_x→a f(x) = L

Therefore, 

lim_x→a g(x) = L

⇒ lim_x→∞ f(x) = lim_x→∞ -1/x = 0 and  

⇒ lim_x→∞ h(x) = lim_x→∞ 1/x = 0

Therefore, 

lim_x→∞ g(x) = 0 

Example 2: Find lim_x→π/2 cot(x)

Solution:

As we know cot(x) is continuous and defined at π/2

Therefore,   

lim_x→π/2 cot(x) = cot(π/2) = 0  

Example 3: Find lim_x→π sec(x)

Solution:   

As we know tan(x) is continuous and defined at π

Therefore,

lim_x→π cosec(x) = sec(π) = -1  

Example 4: Find lim_x→π/2 cosec(x)

Solution:   

As we know tan(x) is continuous and defined at π/2

Therefore,   

lim_x→π/2 cosec(x) = cosec(π/2) = 1  

Example 5: Find lim_x→πtan(x)

Solution:   

As we know tan(x) is continuous and defined at π

Therefore,  

lim_x→π tan(x) = tan(π) = 0   

Example 6: Find lim_x→π cos(x)

Solution:   

As we know cos(x) is continuous and defined at π

Therefore,  

lim_x→π cos(x) = cos(π) = -1 

Example 7: Find lim_x→πsin(x)

Solution:

As we know sin(x) is defined at π

Therefore,   

lim_x→π sin(x) = sin(π) = 0 

Trigonometry is one of the most important branch of Mathematics. We know that there are six trigonometric functions and the limit of trigonometric is the limit taken to each trigonometric function. We can easily find the limit of trigonometric functions and the limit of the trigonometric function may or may not exist depending upon the given function with the point of consideration.

For trigonometric functions, we can easily take the limit of all six trigonometric functions by replacing the function variable with the limit value. The limit of trigonometric function depends on the domain, and range of the function. In this article, we will learn about the limit of all six trigonometric function, their examples and others in detail.