Watch the video below to learn about the Squeeze and L’Hopitals theorem.

- Squeeze Theorem – If you have three functions such that $f(x) \leq g(x) \leq h(x)$ then $\lim f(x) \leq \lim g(x) \leq \lim h(x)$
- L’Hostpital’s Theorem – if $\lim \frac{f(x)}{g(x)} = \frac{0}{0} or \frac{\inf}{\inf}$ then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$

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Now there are times where a limit seems very difficult or even impossible to find a value for directly. This is where the squeeze theorem helps. Instead of directly evaluating the limit given to us, we can find two other limits, one always greater than and one always lesser than the limit we are evaluating. This allows us to ‘squeeze’ the limit between two others, giving us an upper and lower bound on what the value of our limit will be. However if the greater and lesser limits are equal to one another at the point we are evaluating the limit at then the limit in the middle must also have the same value. This allows us to evaluate limits such as $ \frac{\sin{x}}{x}$ as $ x$ approaches $ 0$, by squeezing it between the graphs of $y = 1$ and $y = \cos{x}$. Both of these evaluate to $ 1$ at $ x = 0$ and thus since $ \frac{\sin{x}}{x}$ is between the two $ \frac{\sin{x}}{x}$ must approach $ 1$ at $ x = 0$ as well. Another way of evaluating seemingly impossible limits is when evaluating the limit conventionally gives you a value of $ \frac{0}{0}$ or $ \frac{\infty}{\infty}$. These two situations allow us to use L’Hospital’s rule which states that if a limit can be evaluated to $ \frac{0}{0}$ or $ \frac{\infty}{\infty}$ you can get the actual value of the limit by taking the derivative of the numerator and denominator of the fraction and putting those values as the numerator and denominator of a new limit approaching the same value. If we did this with $ \frac{x^2}{x}$ as $ x$ approaches $ 0$ we would get a new limit of $ \frac{2x}{1}$ as $ x$ approaches $ 0$ which evaluates easily to $ 0$.

Remember to take the quiz before moving on!

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