# 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.

### Notes

These notes from Paul’s Online Math Notes show how to compute various limits. The Squeeze/Sandwich Theorem is discussed from page 6 onwards.

A pdf of this document can be downloaded here.

### 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

### 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.