Sandwich Theorem

The Sandwich Theorem is a theorem that is used to solve limits that would otherwise be considered undefined. It works on the principle that if a function is ‘sandwiched’ between two other functions for a period, the value it converges (i.e. the limit) will also between the limits of the two functions.

It also uses a similar principle to finding the average big-O time of an algorithm.

Definition §

Let $f(x),g(x),h(x)$ all be real-valued functions that operate over a domain of x, where $x\ne a$:

$Ifg(x)\le f(x)\le h(x)$ $and{\lim }_{x\to a}g(x)=L={\lim }_{x\to a}h(x)$ $then{\lim }_{x\to a}f(x)={\lim }_{x\to a}h(x)={\lim }_{x\to a}g(x)=L$ In other words, if f(x) is strictly bound between g(x) and h(x) over a domain and both limits of g(x) and h(x) exist and are equal, then the limit of f(x) is equal to the limits of g(x) and h(x).

Examples §

The trig functions sin and cos are a godsend here, as they are always bound between [-1,1] so any function multiplied by them can be used as g(x) and h(x):

Example - Trig Function

Let $f(x)=x\sin (\frac{1}{x})$. Evaluate ${\lim }_{x\to 0}f(x)$

Solution $\sin (\frac{1}{x})$ is always bounded between -1 and 1, we can use it's domain to determine an upper and lower bound function: $sin(\frac{1}{x})\in [1,-1]$ $⟹|x|\sin (\frac{1}{x})\in [-x,x]$ $⟹-|x|\le f(x)\le |x||x\ne 0$ Since |x| and -|x| are defined to be upper and lower bound functions, and their limits exist and are equal at 0, trivially, then using the Sandwich Theorem:

Since

${\lim }_{x\to 0}f(x)={\lim }_{x\to 0}|x|={\lim }_{x\to 0}-|x|=L=0$ Hence, ${\lim }_{x\to 0}f(x)=0$

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Example - Inverse Functions

Resources §