Calculus
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 26 (Real functions)

Prerequisites: trigonometry, algebra

# Getting Oriented

 Rough Guides to Analysis
 Calculus Real Analysis Complex Analysis

Calculus is the study of change…

# The Basics

## Limits

Our first concept is that of a limit. Limits can be used to evaluate a function at a point where it cannot normally be evaluated.

Definition

The limit of a function f(x) as x approaches a exists and is equal to b if for all $\epsilon>0$ there exists a $\delta>0$ such that whenever $|x-a|<\delta$ then $|f(x)-b|<\epsilon$. Notationally, one writes $\lim_{x\to a) f(x)=b$.

This is sometimes called the epsilon-delta definition of limits. Intuitively, it means that whenever x is close to a, f(x) is also close to b.

A simple example of the need for limits occurs with the function $f(x)=\frac{x^2-1}{x-1}$. Clearly, $f(0)=\frac{0}{0}$ is technically undefined. However, using the fact that $x^2-1=(x+1)(x-1)$, we see that

(1)
\begin{align} \lim_{x\to 1} \frac{x^2-1}{x-1} = \lim_{x\to 1} \frac{(x-1)(x+1)}{(x-1)} = \lim_{x\to 1} (x+1) = 2. \end{align}

Note that we had to keep the $\lim$ on the function until we were able to evaluate it, because technically, the function “inside” the limit was undefined at $x=1$ until then.

## Slopes

One instance in which a limit always occurs is in finding the slope at an arbitrary point of a graph. The slopes of linear functions are easy to find, but this might not be so easy for other functions; one requires the formula

(2)
\begin{align} m_a=\lim_{x\to a} \frac{f(x)-f(a)}{x-a}. \end{align}

As for linear functions, $m$ denotes the slope, but here we use $m_a$ to denote the slope at the specific point $x=a$. As an example, if we're looking for the slope at the graph $x^2$ at $x=1$, then we have

(3)
\begin{align} m_1=\lim_{x\to 1}\frac{x^2-1^2}{x-1}, \end{align}

which is just the limit we computed above and therefore equals 2.

If we draw the line of slope 2 through the point $(1,1)$ on the graph of $f(x)=x^2$, we obtain what is called the tangent line. In general, a tangent line on a graph is easy to draw. Imagine the graph as a physical object and place a board on it at the given point. It is analogous to lines tangent to a circle: the board usually touches the line at a single point.

We can find the equation of the tangent line using the point-slope formula for a line. The slope is $m_a$, of course, given by the limit, and the point of intersection is just $(a,f(a))$, since that is the point on the graph above $x=a$ in Cartesian coordinates. Thus, we have the equation

(4)
$$y-f(a)=m_a(x-a).$$

Rearranging this to $y=mx+b$ form we obtain

(5)
$$y=m_a x + (f(a)-m_a a).$$

Let's check that the tangent line to a given line is always the line itself. If $f(x)=mx+b$, then we have

(6)
\begin{align} m_a=\lim_{x\to a} \tfrac{f(x)-f(a)}{x-a}=\lim_{x\to a}\tfrac{mx+b-(ma+b)}{x-a} =\lim_{x\to a} \tfrac{mx-ma}{x-a} =\lim_{x\to a}\tfrac{m(x-a)}{(x-a)} =\lim_{x\to a}m =m. \end{align}

Using the formula, we have $y=m_a x + (f(a)-m_a a)=m x + (ma+b) - (ma) = mx+b$, as we expected.

# Computing Limits

There are a few ways to compute a limit (listed in approximate order from hardest to easiest).

1. Use the epsilon-delta definition;
2. Reduce a function to include a limit whose value is known;
3. Reduce a function to a point at which the value $a$ may be directly plugged in;
4. Plot the function on a calculator and read off an approximate value using the TRACE function.

In the example above, we used the third method. Generally a combination of the second two is used. The first is only used when a strict mathematical proof is required, while the fourth is only able to give an approximate answer.