Limits Examples

Example 1

The following example comes from calculusforyou (swipe to see the solution to Q275)

If we directly evaluate the above expression, we see that as t tends to 0, we are presented with an indeterminate form. We can solve this limit by applying l’Hopital’s rule (exercise left up to the reader), which consists of calculating the derivative of both the numerator and the denominator separately. We then get

If possible, it is always good to graph your equation to see how it behaves.

Example 2

The following example solution comes from calculusforyou (swipe to see the solution to Q257)

Using the Taylor expansion for cos(x)

If possible, it is always good to graph your equation to see how it behaves, i.e., a straight line where y = 1/384 for all x.

Additional Reading

“Indeterminate Form – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Indeterminate_form.

“L’Hopital’s Rule – Calculus Tutorials”. 2022. math.hmc.edu. https://math.hmc.edu/calculus/hmc-mathematics-calculus-online-tutorials/single-variable-calculus/lhopitals-rule/.

“L’Hôpital’s Rule – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule.

“Taylor Series”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/taylor-series.html.

“Taylor Series – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Taylor_series.