Need for limits

Limits help us understand the behavior of a function at points even when its not explicitly defined. \

Activity

Consider the function $f(x)=e^{\frac {-1}{x^{2}}}$

We want to know about the behavior of function at $x=0$however $f(0)$is not defined since ${\frac {1}{0}}$can't be operated upon. To solve this dilemma we will look at the behavior of $f(x)$when x is near to 0.

In a spreadsheet, plot the values of f(x) as x is the 'neighborhood' of 0. Then plot the function and mark your observations.

Solution

Download this geogebra file from this link.

We can see that as the values approach 0, $f(x)$get really really small and really close to 0 however at no point, does it touch 0. And as the values go away from 0 $f(x)$starts getting bigger.
Download this geogebra file from this link.

The same can be observed from the plot.