Limits Practice Problems

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

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

Lim_x→p f(x) = L

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

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 {ln(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) = 0

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, Continuity, and Differentiability
Limit Formula