Limit of a function: Definition, Types, Properties, and Calculations

In 1861, first-time mathematician Karl Weierstrass defined a limit by using the epsilon-delta. To represent the limit of the function, he used the symbol lim. In 1908, Hardy was the first to use the arrow sign below the limit in the book “A Course of Pure Mathematics”.

Derivation, integration, and continuity is some important application of limit. We can find derivatives by using a limit. We use limits in integrals where upper and lower limits are given. Limits are used to determine whether the function is continuous at any point.

In this article, we will discuss the definition of the limit with its mathematical representation. We will read about some important properties of limits. Further, we will learn how to calculate the limit of any function through some examples.

Definition of limit

Limit is described as the value that the function approaches output for the given value. Suppose f(x) is a real function, x approaches any real number c, and a function f(x) approaches a specific number L, then L is called the limit of a given function. Mathematically limit is written as,

lim_x→c f(x) = L

Different types of limit

Before discussing the types of limits, we should know about the left-hand side limit and right-hand side limit. In the left-hand side limit, x approaches c from the left side, and x is quite close to c but less than c, and it is denoted by lim_x→c^– f(x) = L. Right-hand side limit is written as lim_x→c⁺f(x) = L, where x approaches c from the right side, in this case, x > c.

We will discuss the following four different types of limits.

• One-sided limit
• Two-sided limit
• Infinite limit
• Limit at infinity

Let’s discuss the above types of limits in detail one by one.

1.     One-sided limit

Only one side limit (left or right-hand side) exists in a one-sided limit.

2.     Two-sided limit

In the two-sided limit, limits exist from both sides the Left and right.

3.     Infinite limit

The infinite limit function gives output only +∞ or -∞. Or we can say the limit increase or decrease to infinity.

4.     Limit at infinity

When we want to determine, how much function increase by increasing x. in this case x approaches +∞ or – ∞

Properties of the limit of the function

The properties of the limit of the function are described below:

Let f and g be two function for which lim _x→c f(x) = L and lim _x→c f(x) = M, then

• lim _x→c [k f(x)] = k lim _x→c [f(x)] = k L (Here k is any non-zero constant)
• lim _x→c [f(x) + g(x)] = lim _x→c [f(x)] + lim _x→c [g(x)] = L + M
• lim _{x→ c} [f(x) – g(x)] = lim _x→c [f(x)] – lim _x→c [g(x)] = L – M
• lim _x→c [f(x).g(x)] = lim _x→c[f(x)]. lim _x→c [g(x)] = L.M
• lim _x→c [f(x) / g(x)] = lim _x→c [f(x)] / lim _x→c [g(x)] (Here limit of g(x) should be non-zero)

Some important results

Limit of some important functions are given below

• lim _{x→ a} [x ⁿ – aⁿ /x – a] = n a^n-1
• lim _{x→ +∞} (1 + 1 / n)ⁿ = e
• lim _{x→ 0} (a^x – 1) / x = log_e^a
• lim _{θ→ 0} (Sin θ / θ) = 1
• lim _{x→ ∞} (e^x) = ∞
• lim _{x→ – ∞} (e^x) = 0
• lim _{x→ ± ∞} (a / x) = 0 (where “a” is any real number)
• lim _{θ→ 0} (tan θ / θ) = 1
• lim _{θ → 0} Cos θ  = 1
• lim _{x → ∞} (1 + 1 / n )ⁿ = e

Examples with a step-by-step solution

Here are some examples to find the limit of a function at specific point.

Example 1:

Calculate the limit of the given function 5x³ + x² + x when x approaches 2.

Solution:

Step 1: By using the definition of limit i.e. lim_x→c f(x)

In the given example f(x) = 5x³ + x² + x, and c = 2

Step 2: Put the given data in lim_x→c f(x), then we get,

lim_x→2 f(x) = lim_x→2 (5x³ + x² + x)

Step 3: By using the property of limit, we can write

lim_x→2 f(x) = lim_x→2 (5x³ + x² + x) = lim_x→2 (5x³) + lim_x→2 (x²)+ lim_x→2 (x)

Step 4: Again using the limit property, write the constant outside the limit.

lim_x→2 f(x) = 5 lim_x→2 (x³) + lim_x→2 (x²)+ lim_x→2 (x)

Step 5: After applying the limit, we get

lim_x→2 f(x) = 5 (2³) + (2²) + (2)

Step 5: Simplify the above expression

lim_x→2 f(x) = 5 (8) + (4) + (2) = 40 + 4 + 2

lim_x→2 (5x³ + x² + x) = 46

Example 2:

Simplify

Lim _x→3 (x² – 9) / (x – 3)

Solution:

If we apply the limit we get the (0 / 0) form.

As we know that (a² – b²) = (a – b) (a + b) applying it in the given example.

Lim _x→3 (x² – 3²) / (x – 3)

Lim _x→3 (x – 3) (x + 3) / (x – 3)

(x – 3) will be canceled, we have

Lim _x→3 (x + 3)

Lim _x→3 (x) + Lim _x→3 ­(3) = 3 + 3 =6

So,

Lim _x→3 (x² – 9) / (x – 3) = 6

Conclusion

In this article, we read about the importance of the limit of the function. We described the definition of the limit. Then, we discussed four different types of limits. Further, we defined some properties of limit. After this, we discussed the limit of some important functions.