# Introduction to Limits

### What is a Limit?

### Video

This video from *Khan Academ*y gives a graphical explanation of the idea behind a limit, and why it is more complicated than just substitution.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from Maynooth University introduce the idea of a limit and show the difference between computing a limit and direct substitution.

A pdf of this document can be downloaded here.

### Finding Limits Using Direct Substitution

### Video

This video from *Khan Academy* shows how to evaluate a limit using direct substitution.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

### Notes

These notes from Maynooth University show how to compute limits by direct substitution. There are exercises at the end for you to practise with.

A pdf of this document can be downloaded here.

### Interactive Exercises

You can use this NUMBAS exercise to practise evaluating limits by direct substitution.

Please click here to view this NUMBAS question in another window.

### Finding Limits Using Factorisation

### Videos

This video from *Krista King* shows how to use factorisation of algebraic expressions to evaluate the limit of a rational function when direct substitution results in an indeterminate form.

Note: All *Krista King* content is available for free at www.youtube.com/c/Integralcalc.

Krista King’s Website: www.kristakingmath.com .

### Notes

These notes from Maynooth University show how to compute limits of rational functions by factorisation in the case where direct substitution results in an indeterminate form.

A pdf of this document can be downloaded here.

### Interactive Exercises

### Finding Limits Using Conjugates/Rationalisation

### Videos

This video from *Krista King* shows how to compute limits where the conjugate method is required.

Note: All *Krista King* content is available for free at www.youtube.com/c/Integralcalc.

Krista King’s Website: www.kristakingmath.com .

### Notes

These notes from Maynooth University show how to use rationalisation of algebraic expressions involving square roots to compute limits that result in indeterminate forms when direct substitution is used, and when factorisation does not work.

A pdf of this document can be downloaded here.

### Interactive Exercises

This NUMBAS exercise lets you practise using conjugates of terms involving square roots to compute certain types of limits.

Please click here to view this NUMBAS question in another window.

### Which Technique do I use to Find a Limit?

### Videos

This video from *Khan Academy* goes through a flowchart to help you familiarise yourself with which techniques are appropriate for finding limits given a certain question.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.