# More Advanced Limits

## The Squeeze/Sandwich Theorem

## Video

This video from *PatrickJMT* shows how to use the Squeeze/Sandwich Theorem to find the limit of a function which is between two other functions on an interval.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

This second video from *PatrickJMT* uses the Squeeze/Sandwich Theorem to deal with an example where the bounds of the function are not explicitly given.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

## Limits of Certain Functions Involving the Sine Function

## Video

This video from *Khan Academy* gives the reasoning for why $\displaystyle\lim_{x \rightarrow 0}\dfrac{\sin(x)}{x}=1$, using the unit circle and the Squeeze/Sandwich Theorem.

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

## Notes

These notes from *Maynooth University* show how to compute limits of functions involving the sine function and linear algebraic terms, e.g. $2x$.

A pdf of this document can be downloaded here.

## Interactive Exercises

This NUMBAS exercise lets you practise computing limits of functions of the form $\dfrac{\sin(ax)}{bx}$.

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

## Limits at Infinity

## Videos

This video from *patrickJMT* shows how to compute the limit as the variable tends towards infinity of a rational function. The three cases of the relationship between the degrees of the numerator and denominator are discussed.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

## Notes

These notes from Maynooth University have worked examples of computing limits of rational functions as $x\rightarrow\infty$.

A pdf of this document can be downloaded here.

## Interactive Exercises

This NUMBAS exercise lets you practise computing limits of rational functions as $x$ tends to infinity.

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

## One-Sided Limits

## Videos

These videos from *PatrickJMT* show how to compute the left- and right-hand limits for a piecewise-defined function.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

## Notes

These notes from Paul’s Online Notes introduce the idea of one-sided limits and show how to compute them using both the definition and the graph of a piecewise-defined function.

A pdf of this document can be downloaded here.

## Interactive Exercises

You can practise computing one-sided limits for piecewise-defined functions with this NUMBAS exercise.

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

## Indeterminate Limits and L’Hôpital’s Rule

## Videos

This video from *Khan Academy* introduces L’Hôpital’s Rule and finishes with an introductory example for the indeterminate form $\tfrac{0}{0}$.

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

This video from *Khan Academy* shows how to use L’Hôpital’s Rule for limits of rational functions where substitution results in the indeterminate form $\pm\tfrac{\infty}{\infty}$.

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

This video from *Khan Academy* discusses repeated uses of L’Hôpital’s Rule for the case when applying it once also produces an indeterminate form.

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

## Notes

These notes from Paul’s Online Notes explain L’Hôpital’s Rule and work through various example questions.

A pdf of this document can be downloaded here.

## Interactive Exercises

You can use these NUMBAS exercises to practise using L’Hôpital’s rule to compute limits where direct substitution results in an indeterminate form.

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

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

## Continuity of Functions

## Video

This video from *Khan Academy* explains the idea of the continuity of a function.

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

These videos from *Khan Academy* show how to test a piecewise-defined function for continuity. The first example is continuous and the second is discontinuous.

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

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

## Notes

These notes from Paul’s Online Notes define continuity of functions and show how this concept relates to one-sided limits. Several examples are worked through, and the Intermediate Value Theorem is also discussed.

A pdf of this document can be downloaded here.

## Interactive Exercises

You can use this NUMBAS exercise to practise finding discontinuities in piecewise-defined functions.

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