Introduction to Limits – Limits and Continuity

Introduction to Limits

The limit of a function is concerned with the behaviour of a function near a given point. What happens at the point is of no concern. To be more precise,

if “the limit of the function $$f(x)$$ as $$x$$ approaches the point $$a$$” is the value $$L$$ then this is denoted as$\lim_{x\to a}f(x) = L.$

Consider the graph to the right where$f(x) = \left\{ \begin{array}{ll} 3, & x \ne 4 \\ 5, & x = 4. \end{array} \right.$What is $$\displaystyle\lim_{x\to 4}f(x)$$? There are two ways to approach $$x=4$$. From the left and from the right. From the left the limit is $$3$$ and from the right the limit is $$3$$. Since these limits from the left and right are the same, we can say that$\lim_{x\to 4}f(x) = 3.$

Note that the limit has nothing to do with what happens at $$x=4$$ since $$f(4) = 5$$ which is not equal to $$\displaystyle\lim_{x\to 4}f(x) = 3$$.

Graphical Way of Looking at Limits

Now consider the graph to the left. What is $$\displaystyle\lim_{x \to 3} f(x)$$? One can approach this point two ways. From the left, $$\displaystyle\lim_{x \to 3^-} f(x) = 1$$ and from the right $$\displaystyle\lim_{x \to 3^+} f(x) = 4$$.

In this example the limit does not exist (DNE) because the limit we get as we approach $$x=3$$ depends on the direction in which we approach $$x=3$$.

Because of this problem of obtaining a different limit based on the direction in which we approach a point we need the concept of left and right hand limits.

Right and Left Hand Limits

Notice that $$\lim_{x \to 0} f(x)$$ DNE. Therefore, the limit you get depends on how you approach $$x = 0$$.

As you approach from the left (also called from below’) $$\lim_{x \to 0^-} f(x) = 4 = L^-$$.

As you approach from the right (also called from above’) $$\displaystyle \lim_{x \to 0^+} f(x) = -3 = L^+$$.

In order for the limit to exist at a point, we first require that both the left and right hand limits must exist. By this we mean that both $$L^+$$ and $$L^-$$ are finite numbers. Second, we require that these values are the same, $$L^+ = L^-$$.

We can summarize these ideas in the following theorem.
Theorem 1.1 We say that the limit of $$f(x)$$ as $$x$$ approaches $$a$$ exists and equals $$L$$, denoted as $$\displaystyle\lim_{x\to a} f(x) = L$$
1. if $$\displaystyle \lim_{x \to a^-} f(x) = L^-$$ exists,
2. $$\displaystyle \lim_{x \to a^+} f(x) = L^+$$ exists,
3. and $$L^+ = L^- = L$$.

Note that both $$L^+$$ and $$L^-$$ must exist and not be $$\pm\infty$$.