# MAT 141, Day 9, 2017

## MAT 141 Day 9, 2017

Here is an interactive graphic to see see how the squeeze theorem works.

@interact def _(f =x, x0=(.5,(-1,1)),show_outside=False): pretty_print(html("We will multiply $\\sin(1/x)$ by your function")) f(x) = f pt = point((x0,f(x0)*sin(1/x0)),color='red',pointsize=50,zorder=5) pl = plot(f(x)*sin(1/x),(x,-1,1),linestyle='--',zorder=1) p2 = plot([f,-f],(x,-1,1),color='green') if show_outside: show(pl+pt+p2,figsize=4) else: show(pl+pt,figsize=4) pretty_print(html("This is $%s\\sin(1/x)$"%latex(f(x))))

## Click to the left again to hide and once more to show the dynamic interactive window

Sage can calculate many (but not all) limits for you.   Click "evaluate" to try them out!

limit?
limit(sin(1/x),x=0)
 ind ind

We can have directional limits.

limit(1/x,x=0,dir='left')
 -Infinity -Infinity
limit(2/x-2/abs(x),x=0,dir='left')
 -Infinity -Infinity
limit(1/x^2,x=0,dir='below')
 +Infinity +Infinity
limit(x^2+3,x=0)
 3 3
lim(ln(abs(1/x)),x=0)
 +Infinity +Infinity

Here's an example showing that Sage can do some squeeze theorem examples as well.

limit(x^2*cos(1/x),x=0)
 0 0

## Homework

Use this cell to try computing $\lim_{x\to 1}\frac{x-1}{\ln(x)}$ for yourself!