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)
\begin{equation} y-f(a)=m_a(x-a). \end{equation}

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

(5)
\begin{equation} y=m_a x + (f(a)-m_a a). \end{equation}

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.

Rules for Differentiation

Going Further

The Road Ahead

This article is concerned entirely with single-variable functions, those with one input and one output. The next step in calculus is to extend the ideas of differentiation and integration to functions with more than one input or output. This is covered in multivariable calculus.

A more rigorous treatment of calculus is typically given in a first course on real analysis. In that course, the details behind the definition of limits, derivatives, and integrals are examined in much more depth.

References

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