Properties of Limits

In mathematics, a limit is a concept that describes how a function or sequence behaves as its input gets closer to a certain value. For example, if you keep dividing 1 by larger and larger numbers (like 1/2, 1/3, 1/4, etc.), the result gets closer and closer to 0. The limit of this sequence as the denominator gets infinitely large is 0.

Properties of limits tell us how limits behave when we perform different operations on functions. These properties make it easier to calculate limits by breaking down complex expressions into simpler parts. By using these rules, we can add, subtract, multiply, and divide functions while still working with their limits.

Algebra of Limits

Let's say we have two functions, f(x) and g(x). We know that \lim_{x \to a}f(x) and  \lim_{x \to a}g(x) exist then some of the algebra of these limits is discussed below: 

Sum of Limits

 The limit of the sum of two functions is the sum of the limits of both functions.

\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)

Difference of Limits

 The limit of the difference of two functions is the difference of the limits of both functions. 

\lim_{x \to a}[f(x) - g(x)] = \lim_{x \to a}f(x) - \lim_{x \to a}g(x)      

Product of Limits

 The limit of the product of two functions is the product of the limits of both functions. 

\lim_{x \to a}[f(x).g(x)] = \lim_{x \to a}f(x). \lim_{x \to a}g(x)      

Quotient of Limits

 The limit of the quotient of two functions is the quotient of limits of both functions. 

\lim_{x \to a}[\dfrac{f(x)}{g(x)}] = \dfrac{\lim_{x \to a}f(x)}{ \lim_{x \to a}g(x)}

Constant Multiple

If f(x) has a limit as x → c, then:

\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)

Where k is constant.

Power of a Function

\lim_{x \to c} \left[ f(x) \right]^n = \left( \lim_{x \to c} f(x) \right)^n

for any integer n.

Function as Exponent

\lim _{x \rightarrow a}[f(x)]^{g(x)} = \left[\lim _{x \rightarrow a} f(x)\right]^{\lim _{x \rightarrow a} g(x)}

Limit as Constant

\lim _{x \rightarrow a} \ c = c, c is any real number

Identity Property

\lim _{x \rightarrow a} \ x = a

Power Property

\lim _{x \rightarrow a} \ x^n = a^n

Limit of Composite Functions 

The composition of two functions f(x) and g(x) is denoted by (f o g)(x), which means the range of the function g(x) should lie in the domain of the function f(x). Now for calculating the limit of the composition of the two functions, we use the following property: 

\lim_{x \to a}(f o g)(x) = \lim_{x \to a}f(g(x)) = f(\lim_{x \to a}g(x))      

Summary

These all properties can be summarized as:

Read More,

• Limit Formulas
• How to Find Limits?
• L’ Hopital Rule

Solved Examples on Properties of Limits

Example 1: Given the function f(x) = \frac{1}{x^2}. Find \lim_{x \to 0}\frac{1}{x^2}. 

Solution: 

Let's see this limit graphically, 

We can see from the graph while approaching the function from either of the sides towards zero. Values start going to infinity. 

\lim_{x \to 0^-}f(x) = \lim_{x \to 0^+}f(x) = \infty 

Example 2: Find the value of the limit of the function f(x) = x + cos(x) when x ⇢ 0.

Solution: 

The figure below shows the graph of the function, 

We know that f(x) is a combination of two different function. We can use the properties studied above, property 1 works for our case.

\lim_{x \to a}[f(x) + g(x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)

We know f(x) = x + cos(x). Let's say h(x) = x and g(x) = cos(x) and using the above property we get. 

\lim_{x \to 0}[h(x) + g(x)] = \lim_{x \to 0}h(x) + \lim_{x \to 0}g(x)

= \lim_{x \to 0}x + \lim_{x \to 0}cos(x)

= 0 + 1

= 1

Example 3: Find the value of the limit of the function f(x) = (x² + x +1)e^x when x ⇢ 0.

Solution: 

We know that f(x) is a combination of two different function. We can use the properties studied above, property 3 works for our case.

\lim_{x \to a}[f(x).g(x)] = \lim_{x \to a}f(x). \lim_{x \to a}g(x)

We know f(x) = (x² + x +1)e^x Let's say h(x) = x² + x +1 and g(x) =e^x and using the above property we get. 

\lim_{x \to 0}[h(x).g(x)] =( \lim_{x \to 0}h(x))(\lim_{x \to 0}g(x))

= \lim_{x \to 0}x^2 + x + 1 \times \lim_{x \to 0}e^x

= 1 + 1

= 1

Example 4: Find the value of the limit of the function f(x) = \frac{cos(x)}{x^2 + x + 4} when x ⇢ 0.

Solution: 

We know that f(x) is a combination of two different function. We can use the properties studied above, property 4 works for our case.

\lim_{x \to a}[\frac{f(x)}{g(x)}] = \frac{\lim_{x \to a}f(x)}{ \lim_{x \to a}g(x)}      

We know f(x) = \frac{cos(x)}{x^2 + x + 4}      Let's say g(x) = x² + x +4 and g(x) =cos(x) and using the above property we get. 

\lim_{x \to a}[\frac{f(x)}{g(x)}] = \frac{\lim_{x \to a}f(x)}{ \lim_{x \to a}g(x)}

= \frac{\lim_{x \to 0}cos(x)}{\lim_{x \to 0}x^2 + x + 4} 

= \frac{cos(0)}{0 + 0 + 4}

= \frac{1}{4}

Example 5: Find the value of the limit of the function from left-hand side and right-hand side when x ⇢ 0, f(x) = f(x)= \begin{cases}    3,& \text{if } x \geq 0\\  -1,& \text{otherwise} \end{cases}   .

Solution: 

Let's see this limit graphically, 

Notice in the graph that while approaching from the left-hand side, the functions seems to take value -1 and while approaching from the right-hand side, functions seems to taking value 3. 

Thus, 

\lim_{x \to 0^-}f(x) = -1

\lim_{x \to 0^+}f(x) = 3

Practice Questions on Limits

Question 1: Given the function f(x) = \frac{1}{x^3}, \quad \text{find} \ \lim_{x \to 0} \frac{1}{x^3}.

Question 2: Find the value of the limit of the function f(x) = x + tan⁡(x) when x→0.

Question 3: Find the value of the limit of the function f(x) = (x² + 1)e^x when x→0.

Question 4: Find the value of the limit of the function f(x) = \frac{\cos(x)}{x^2 + 2x + 5}, \quad \lim_{x \to 0} \frac{\cos(x)}{x^2 + 2x + 5}

Question 5: Find the value of the limit of the function from the left-hand side and right-hand side when x→0: f(x) = \begin{cases} 5, & \text{if } x \geq 0 \\ -3, & \text {otherwise} \end{cases}