Intro to Limits

Limits are the basis for much of calculus. They are also highly abstract. The development of calculus was stalled partially due to the inability of mathematicians to deal with the limits. Limits are tied closely to the idea of infinity (both infinitely large and small).

Limits, informally, are the concept of investigating what happens to a function as its argument gets closer and closer to a particular value. Most functions you have seen behave nicely (and thus are rather boring) but a few behave differently… But before we jump into functions lets look at sequences on numbers. Below is a sequence of numbers that approaches a value as the "term number" gets larger:

Term Number $n=1$ $n=2$ $n=3$ $n=4$ $n=5$
Value of the Term 0.3 0.33 0.333 0.3333 0.33333

As the sequence progresses the value of the term gets larger, but the value of each successive term is getting closer to a value. Think about that… The number is getting larger and larger but there is a limit on how big that number can get (if we follow the pattern). No matter how many terms you choose to look at the value of the term simply gets closer and closer to $\frac{1}{3}$!

Limits of function are very similar. Lets start with the function $h(x)=2^{-x}$. Lets start by good old fashioned plug-and-chug, first looking at large positive values of x and see what happens to the function:

Value of x 2 5 10 15
Value of the Function 0.25 0.03125 0.000977 0.0000305

So what is happening to the function as x gets larger and larger? Clearly the function is approaching zero. This can be seen easily (and more quickly) by simply graphing the function with your calculator. In fact many (but not all) limit questions can be resolved by simply graphing the function and observing what happens. The plug-and-chug method can also be used and at times is more accurate…

Limit Notation

Again, in limits we ask the question what happens to a function as a variable does something. The notation for limits looks something like:


The entire notation is the limit, not just the "lim" part.

A Few Examples

Example 1: Find the limit of $g(x)=x^2$ as x approaches 2.

\begin{align} \lim_{x \rightarrow 2} x^2 = ? \end{align}

This can be done graphically or numerically, but in this case numerically might be easier (limits as the argument approaches positive or negative infinity are often most easily done graphically).

Value of x 1 1.5 1.9 1.99
Value of the Function 1 2.82 3.732 3.972

If we continue on it becomes clear, and not surprising, that the function is approaching the value of 4. Thus formally we would write:

\begin{align} \lim_{x \rightarrow \infty} x^2 = 4 \end{align}

Example 2: A bit trickier. Find the limit of $j(x)=\frac{5x+x^2}{x}$ as x approaches 0.

\begin{align} \lim_{x \rightarrow 0} \frac{5x+x^2}{x} = ? \end{align}

This example is not a simple as the previous this function is undefined for $x=0$. Solving this one graphically can fail to shed all the light that needs to be shed. If you could zoom in or your calculator is smart you would see that there is a gap in the graph at $x=0$. So again, we are investigating what happens to the function as it gets close to 0 not what happens at 0! In this case we will need to do a bit of algebra. Both terms in the numerator have the common term of x, so lets factor it out (this is most useful as all, in this case only one, term is a factor of x).

\begin{align} \lim_{x \rightarrow 0} \frac{5x+x^2}{x} = \lim_{x \rightarrow 0} \frac{x(5+x)}{x} = \lim_{x \rightarrow 0} \: (5+x) = 5 \end{align}

With a little algebra, the problem has become far more tangible. It is fairly clear to see that as x gets closer and closer to 0 that the function will get closer and closer to 5. This problem could have been tackled numerically with a bit of plug and chug from the very beginning, but the algebra is a bit more elegant and frankly easier and faster.

Rules for Limits

I provide these as a reference. If I find the time I might give a few proofs by example…

\begin{align} \lim_{x \rightarrow 0} \: c = c \end{align}
\begin{align} \lim_{x \rightarrow a} \: c \: f(x)= c \: \lim_{x \rightarrow a} f(x) \end{align}
\begin{align} \lim_{x \rightarrow a} [f(x) \pm g(x)] = \lim_{x \rightarrow a} f(x) \pm \lim_{x \rightarrow a} g(x) \end{align}
\begin{align} \lim_{x \rightarrow a} [f(x) \times g(x)] = \lim_{x \rightarrow a} f(x) \times \lim_{x \rightarrow a} g(x) \end{align}
\begin{align} \lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{ \lim_{x \rightarrow a} f(x)} { \lim_{x \rightarrow a} g(x)} \end{align}

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