6. Squeeze Theorem

Mass-Spring-Damper

$x(t)=e^{-\frac{t}{10}}(\cos (t)+\frac{1}{10}\sin (t))$

Looking at the graph above, we can tell that the horizontal asymptote is $y=0$. The wave oscillation is within a so-called envelope by the 2 functions

$-\frac{11}{10}{e}^{-\frac{t}{10}}\le x(t)\le \frac{11}{10}{e}^{-\frac{t}{10}}$

In this envelope, it also says that $y=0$.

• Green is $\frac{11}{10}{e}^{-\frac{t}{10}}$
• Red is $-\frac{11}{10}{e}^{-\frac{t}{10}}$

Squeeze Theorem

Theorem

Suppose $f(x)$, $g(x)$ and $h(x)$ are three functions such that

$g(x)\le f(x)\le h(x)$

for $x$ greater than some value $M$ and suppose

${\lim }_{x\to \infty }g(x)=a={\lim }_{x\to \infty }h(x)$

for some value $a$. Then

${\lim }_{x\to \infty }f(x)=a$