Introduction To Limits

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The concept of limits forms the basis of calculus and is a very powerful one. Both differential and integral calculus are based on this concept and as such, limits need to be studied in good detail.

This section contains a general, intuitive introduction to limits.

Consider a circle of radius r.

We know that the area of this circle is \(\pi {r^2}\) How?

The ancient Greeks derived this result using the concept of limits.

To see how, recall the definition of \(\pi \) .

\(\begin{align}&\pi = \frac{{{\rm{length}}\,{\rm{of}}\,{\rm{circumference}}}}{{{\rm{length}}\,{\rm{of}}\,{\rm{diameter}}}}\\&\pi = \frac{c}{d} = \frac{c}{{2r}} \\&c = 2\pi r\end{align}\)

With this definition in hand, the Greeks divided the circle as follows (like cutting a cake or a pie):

Now they took the different pieces of this ‘pie’ and placed them as follows:

See what happens if the number of cuts are increased

The figure on the right side starts resembling a rectangle as we increase the number of cuts to the circle. The sequence of curves that joins x to y starts becoming more and more of a straight line with the same total length \(\pi r\) .

What happens as we increase the number of cuts indefinitely, or equivalently, we decrease \(\theta \) indefinitely? The figure ‘almost’ becomes a rectangle, though never becoming a rectangle exactly. The area ‘almost’ becomes \(\pi r \times r = \pi {r^2}.\)

In the language of limits, we say that the figure tends to a rectangle or the area A tends to \(\pi {r^2},\) or the limiting value of area is \(\pi {r^2}.\)

In standard terminology.

\[\mathop {{\rm{lim}}}\limits_{\theta  \to 0} {\rm{A}} = \pi {r^2}\]

Hence, we see that a limit describes the behaviour of some quantity that depends on an independent variable, as that independent variable ‘approaches’ or ‘comes close to’ a particular value.

For example, how does \(\begin{align}\frac{1}{x}\end{align}\) behave when x becomes larger and larger? \(\begin{align}\frac{1}{x}\end{align}\) becomes smaller and smaller and ‘tends’ to 0.

We write this as

\[\mathop {\lim }\limits_{x \to \infty } \frac{1}{x} = 0\]

How does  \(\begin{align}\frac{1}{x}\end{align}\) behave when x becomes smaller and smaller and approaches 0? \(\begin{align}\frac{1}{x}\end{align}\) obviously becomes larger and larger and ‘tends’ to infinity.

We write this as:

\[\mathop {\lim }\limits_{x \to 0} \frac{1}{x} = \infty \]

The picture is not yet complete. In the example above, x can ‘approach’ 0 in two ways, either from the left hand side or from the right hand side:

\(x \to {0^ - }\) : approach is from left side of 0

\(x \to {0^ + }\) : approach is from right side of 0

How do we differentiate between the two possible approaches? Consider the graph of \(\begin{align}f\left( x \right) = \frac{1}{x}\end{align}\) carefully.

As we can see in the graph above, as x increase in value or as \(x \to \infty ,\,\,f\left( x \right)\) decreases in value and approaches 0 (but it remains positive, or in other words, it approaches 0 from the positive side)

This can be written

\[\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = {0^ + }\]

Similarly,

   \[\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) =  + \,\infty \]

What if x approaches 0, but from the left hand side \(\left( {x \to {0^ - }} \right)?\) From the graph, we see that as \(\begin{align}x \to {0^ - },\,\,\frac{1}{x}\end{align}\) increases in magnitude but it also has a negative sign, that is \(\begin{align}\frac{1}{x} \to - \,\infty .\end{align}\)

What if \(\begin{align}x \to - \,\infty ?\,\,\,\frac{1}{x}\end{align}\) decreases in magnitude (approaches 0) but it still remains negative, that is, \(\begin{align}\frac{1}{x}\end{align}\) approaches 0 from the negative side or \(\begin{align}\frac{1}{x} \to {0^ - }\end{align}\)

These concepts and results are summarized below: