Introduction to Limits

This video from Khan Academy 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.

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.

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.

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.

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

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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 .

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.

These NUMBAS exercises involve computing limits of algebraic functions using factorisation.

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

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

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 .

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.

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.

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.