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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!

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.