Chapters

Exercise 1
Exercise 2
Exercise 3
Calculate the Following Limits:
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6
Solution of exercise 7

The best Maths tutors available

Exercise 1

Using the definition of a limit, prove that:

$\text{[math]}$

Exercise 2

Using the graph of the function f(x), determine the following limits.

Exercise 3

Using the definition of a limit, prove that:

$\text{[math]}$ has a limit −1 as x 0

Calculate the Following Limits:

Exercise 4

$\text{[math]}$

Exercise 5

$\text{[math]}$

Exercise 6

$\text{[math]}$

Exercise 7

$\text{[math]}$

Solution of exercise 1

Using the definition of a limit, prove that:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

If $\text{[math]}$

$\text{[math]}$

To check this, take a $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$

For $\text{[math]}$ .

Solution of exercise 2

Using the graph of the function f(x), determine the following limits.

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Solution of exercise 3

Using the definition of a limit, prove that:

$\text{[math]}$ has a limit −1 as x 0

$\text{[math]}$

Left side limit.

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Right side limit

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Solution of exercise 4

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

Calculate the side limits to determine the sign of $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$

No limit.

Solution of exercise 5

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

Solution of exercise 6

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

Solution of exercise 7

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Did you like this article? Rate it!

4.00 (4 rating(s))

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

Theory

Continuity on a Closed Interval

Limit of a Logarithmic Function

Limit of an Exponential Function

Limits and Dividing by Zero: Theory and Examples

Formulas

Exercises

Intermediate Value Theorem Problems

Cancel reply

Don

November 2025

Derivative of denominator is wrong

Vanessa

November 2025

Good catch—thanks for pointing that out! We’ll double-check the derivative in that section and make any necessary corrections. Really appreciate you taking the time to flag it. 👍

Abbas Adamu

July 2025

Thank you

Vanessa

August 2025

Thank you Abbas! Good luck with your studies!

Mark Morton

July 2025

With regard to the Zero Over a Number item, is there a mis-statement? It’s immediately followed by “If a number is divided by zero which means that the numerator is zero and the denominator is the number, then the result is zero.”

Vanessa

July 2025

Hi Mark,

You’re absolutely right to raise the question — there does appear to be a misstatement in that sentence. The phrase “If a number is divided by zero, which means that the numerator is zero and the denominator is the number…” is indeed misleading and should be corrected.

To clarify:

Zero divided by a number (e.g. 0 ÷ 5) equals 0.

A number divided by zero (e.g. 5 ÷ 0) is undefined.

We’ll update the sentence to reflect the correct mathematical explanation. We appreciate you catching that and helping us improve the accuracy of the content!

Janice

July 2025

There is more than one size of infinity, though. What if you multiply the infinity of the whole numbers (Aleph-0) by the infinity of the real numbers (fraktur-c)?

Piyash Mia

May 2025

Thanks a lot to you for this essentiol article.

Vanessa

July 2025

Hi Piyash! Thanks for your comment, great to hear that you found this useful!

Jayaraman

May 2025

Very nice

Limit Problems

Exercise 1

Exercise 2

Exercise 3

Calculate the Following Limits:

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Solution of exercise 7

Theory

Zero Over Zero

Calculating Limits

Continuity

Continuous Function

Continuity on a Closed Interval

Discontinuity

Essential Discontinuity

Intermediate Value Theorem

Jump Discontinuity

Limit

One-Sided Limit

Indeterminate Forms

Zero Times Infinity

Infinity Minus Infinity

Infinity over Infinity

Extreme Value Theorem

Bolzano’s Theorem

Limits at Infinity

Infinite Limit

Limit of a Logarithmic Function

Removable Discontinuity

Limit of an Exponential Function

Limit Rules

Properties of Infinity

One to the Power of Infinity

Limits and Dividing by Zero: Theory and Examples

Formulas

Continuity Formulas

Limit Formulas

Exercises

Limits Worksheet

Continuity Problems

Limit Problems

Intermediate Value Theorem Problems

Continuity Worksheet

Cancel reply