Limits Practice Problems

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 (\frac {sinx} {x})= 1

lim_x→0 \frac {log(1+x)} {x} = 1

lim_x→0 log e^x = 0

lim_x→e log x = 1

lim_x→0 \frac {e^x - 1} {x} = 1

lim_x→0 \frac {a^x - 1} {x} = 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, \lim_{{x \to 0}} \frac{\sin x}{x}

Solution:

\lim_{{x \to 0}} \frac{\sin x}{x} = 1

Problem 3: Evaluate lim _x→3 (\frac{x^2 - 9}{x - 3}).

Solution:

Given

\frac{x^2 - 9}{x - 3} = \frac {(x - 3) (x + 3)} {x - 3)}

= x+3

lim _x→3 (x + 3) = 3 + 3 = 6.

Problem 4: Evaluate lim _x→∞ \frac{5x^3 - 2x + 7}{x^3 + 4x^2 + 3}

Solution:

Divide the numerator and the denominator by x³

lim _x→∞ \frac{5 - \frac{2}{x^2} + \frac{7}{x^3}}{1 + \frac{4}{x} + \frac{3}{x^3}}

= 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 \frac {sinx} {x}.

Problem 3: Evaluate lim _x→1 \frac{x^2 - 1}{x - 1}

Problem 4: Evaluate lim _x→∞ \frac{3x^2 + 2x + 1}{4x^2 - x + 5}

Problem 5: Evaluate lim _x→0 e^x - 1.

Problem 6: Evaluate lim _x→3 \frac{1}{x - 3}

Problem 7: Evaluate lim _x→2 \frac{x^2 - 1}{x - 1}.

Problem 8: Evaluate lim _x→3 x - 3.

Problem 9: Evaluate lim _x→0 e^x.

Problem 10: Evaluate lim _x→3 x - 1.