Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity.

What are Limits?

Limits in maths are unique real numbers. Let us consider a real-valued function “f” and the real number “p”, the limit is normally defined as

Lim_x→p f(x) = L

It is read as “the limit of f of x, as x approaches c equals L”. The “lim” shows the limit and the fact that function f(x) approaches the limit L as the right arrow describes x approaches p.

Read More:

• Limits in Calculus: Definition, Formula
• Limits, Continuity, and Differentiability
• Limit Formula

Limits Important Formulas

lim_x→0 (sin x) = 0

lim_x→0 (cos x) = 1

lim_x→0 ([Tex]\frac {sinx} {x}[/Tex])= 1

lim_x→0 [Tex]\frac {log(1+x)} {x}[/Tex] = 1

lim_x→0 log e^x = 1

lim_x→e log x = 1

lim_x→0 [Tex]\frac {e^x – 1} {x}[/Tex] = 1

lim_x→0 [Tex]\frac {a^x – 1} {x}[/Tex] = ln a

Limits: Practice Questions with Solution

Problem 1: Find the value of lim_x→0 x² + 1

Solution:

We have,

lim_x→0 x² + 1

Put x= 0 directly, we get value of limit as 1.

Problem 2: Check for the limit, [Tex]\lim_{{x \to 0}} \frac{\sin x}{x} [/Tex]

Solution:

[Tex]\lim_{{x \to 0}} \frac{\sin x}{x} = 1 [/Tex]

Problem 3: Evaluate lim _x→3 ([Tex]\frac{x^2 – 9}{x – 3} [/Tex]).

Solution:

Given

[Tex]\frac{x^2 – 9}{x – 3}[/Tex] = [Tex]\frac {(x – 3) (x + 3)} {x – 3)}[/Tex]

= x+3

lim _x→3 (x + 3) = 3 + 3 = 6.

Problem 4: Evaluate lim _x→∞ [Tex]\frac{5x^3 – 2x + 7}{x^3 + 4x^2 + 3} [/Tex]

Solution:

Divide the numerator and the denominator by x³

lim _x→∞ [Tex]\frac{5 – \frac{2}{x^2} + \frac{7}{x^3}}{1 + \frac{4}{x} + \frac{3}{x^3}} [/Tex]

= 5 − 0 + 0 / 1 + 0 + 0

= 5

Problem 5: Evaluate lim _x→0 tanx.

Solution:

lim_{x → 0} tan(x) = 1

Problem 6: Evaluate lim_x→2 (8 – 3x + 12x²).

Solution:

lim_x→2 (8 – 3x + 12x²)

= 8 – (3 x 2) + (12 x 4)

= 50

Limits Practice Problems: Unsolved

Problem 1: Evaluate lim_x→2 (3x – 5).

Problem 2: Evaluate lim _x→0 [Tex]\frac {sinx} {x}[/Tex].

Problem 3: Evaluate lim _x→1 [Tex]\frac{x^2 – 1}{x – 1} [/Tex]

Problem 4: Evaluate lim _x→∞ [Tex]\frac{3x^2 + 2x + 1}{4x^2 – x + 5} [/Tex]

Problem 5: Evaluate lim _x→0 e^x – 1.

Problem 6: Evaluate lim _x→3 [Tex]\frac{1}{x – 3} [/Tex]

Problem 7: Evaluate lim _x→2 [Tex]\frac{x^2 – 1}{x – 1} [/Tex].

Problem 8: Evaluate lim _x→3 x – 3.

Problem 9: Evaluate lim _x→0 e^x.

Problem 10: Evaluate lim _x→3 x – 1.

Limits Practice Problems – FAQs

What is a limit in mathematics?

A limit is a fundamental concept in calculus and analysis concerning the behavior of a function as its input approaches a particular point.

How is the limit of a function expressed?

The limit of a function f(x) as x approaches a value a is written as

lim _x→a f(x) = L,

where L is the value that f(x) approaches as x approaches a.

What are some common methods for finding limits?

Some common methods include:

• Direct substitution
• Factoring
• Rationalizing
• Using special limit laws (e.g., limit of a sum, product, quotient)
• L’Hôpital’s Rule for indeterminate forms

What is a one-sided limit?

A one-sided limit is a limit where the function approaches a specific value from only one side (left or right) of a point. It is denoted as

f(x) = lim _x→a- for the left-hand limit and lim_x→a+f(x) for the right-hand limit.

What are indeterminate forms?

Indeterminate forms are expressions where the limit cannot be directly determined and require further analysis.

Why are limits important in calculus?

Limits are essential for defining derivatives and integrals, which are the core concepts of calculus. They help in understanding the behavior of functions, continuity, and the area under curves.