Theorems of Limits of Trigonometric Functions

We have two theorems that are used in defining the limit of the trigonometric functions that are,

Theorem 1

For any two real-valued functions f(x) and g(x) which are defined in the same domain and the relation between them is f(x) ≤ g(x). We can take the limit of these functions at x approaching a then,

lim_x⇢af(x) = f(a) and, lim_x⇢ag(x) = g(a)

If both limits exist then we can easily state that,

lim_x⇢af(x) ≤ lim_x⇢ag(x)

Theorem 2 (Sandwich Theorem)

The theorem is used to calculate the limit of those functions whose limit cannot be calculated easily like (sin x/x at x = 0). The function g(x) is squeezed or sandwiched between two functions h(x) and g(x) in such a way that 

f(x) ≤ g(x) ≤ h(x)

The graph of the above condition is shown below which represents the sandwich theorem limit.

We can say h(x) is the upper limit of g(x) and f(x) is its lower limit at point a as can be observed in the graph above:

lim_x→a h(x) = L and lim_x→a f(x) = L

Where,

• a is the point at which the limit is calculated, and 
• L is the value of the limit.

Then,

lim_x→a g(x) = L

Example: Given: g(x) = x²sin(1/x), Find: lim_x→0 g(x)

Solution: 

We know,

-1≤ sin(1/x) ≤ 1 Under its domain       

Multiplying with x²

-x² ≤ x2sin(1/x) ≤ x²

Then let f(x) = -x² and h(x) = x²

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→0 f(x) = lim_x→0 -x² = 0 and  

⇒ lim_x→0 h(x) = lim_x→0 x² = 0

Thus, lim_x→0 g(x) = 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.