INTRODUCTION TO LIMITS
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Let $f$ be a function—$f$ takes inputs, does something to them, and gives corresponding outputs.

For example, $f$ might take the input $2$ and give the corresponding output $f(2)$.
(Remember: $f(2)$ represents ‘the output from the function $f$ when the input is $2$’.)

Perhaps, however, the function $f$ has become damaged.
Oh dear! It is no longer able to act on the input $2$!
Then, we might find ourselves asking the following question:

When the inputs are close to $2$ (but not equal to $2$),
then what's happening to the corresponding outputs?

This scenario of letting inputs get close to a given number (but never actually getting there),
and exploring the corresponding outputs, leads to the calculus concept of a limit.

Every important calculus idea relies on the idea of a limit!
The idea of a limit is the central idea in calculus!
The purpose of this section is to begin to explore—mostly at an intuitive level—the idea of limits.

a first example: inputs close to $2$, outputs close to $4$

Consider the three sketches below:

However, in all three cases the behavior is the same:
when the inputs are close to $2$ (but not equal to $\,2\,$),
the corresponding outputs are close to $4$

So, how will this behavior be described, using mathematical notation?
In two different (equivalent) ways:

Here's a KEY IDEA:
It doesn't matter what's happening at $\,2\,$!
When we investigate a limit as $\,x\,$ approaches $\,2\,$, we never let $\,x\,$ equal $\,2\,$.
We only let $x$ get arbitrarily close to $\,2\,$.

LIMITS (intuitive) the limit of $f(x)$, as $x$ approaches $c$, equals $\ell$
Consider the mathematical sentence: $$\lim_{x\rightarrow c} f(x) = \ell$$ This sentence is read aloud as:
“ the limit of $\,f\,$ of $\,x\,$, as $\,x\,$ approaches $\,c\,$, equals $\,\ell$ ”


The sentence can alternately be written as $$\text{as } x\rightarrow c,\ \ f(x) \rightarrow \ell$$ in which case it is read aloud as:
“ as $\,x\,$ approaches $\,c\,$, $\,f\,$ of $\,x\,$ approaches $\,\ell$ ”


Under what conditions is this sentence true?
  • (least precise)
    as the inputs get close to $\,c\,$, the corresponding outputs from $\,f\,$ get close to $\,\ell$
  • (more precise)
    we can get the outputs from $\,f\,$ as close to $\,\ell\,$ as desired,
    by requiring that the inputs be sufficiently close to $\,c\,$ (but not equal to $\,c\,$)
  • (most precise; the definition)
    for every $\,\epsilon \gt 0\,$, there exists $\,\delta \gt 0\,$,
    such that if $\,x\in \text{dom}(f)\,$ and $\,0 \lt |x - c| \lt \delta\,$,
    then $\,|f(x) - \ell| \lt \epsilon$

For now, don't worry about that scary-looking definition—a time will come when it will make sense!

OBSERVATIONS ABOUT LIMITS:
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Basic Derivative Shortcuts

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.