Limits and Continuity¶
Limits and continuity are fundamental concepts in calculus that help us describe how functions change as their input values approach certain values or ranges of values. In this article, we will explore these concepts using both intuitive pictures and formal definitions. We will also see examples of computing limits, including one with a removable discontinuity and one with a jump or infinite discontinuity. Finally, we will discuss how limits and continuity are related to each other.
Intuitive Picture of Limits¶
To begin, let's consider what it means for a function to approach a particular value as its input approaches some value \(x=a\). For example, let's look at the function \(f(x) = \frac{1}{x}\). As \(x\) gets closer and closer to 0, the output of the function approaches infinity. We say that this limit is \(\infty\), or more formally, \begin{equation} \lim_{x \to a} f(x) = \infty. \end{equation} In general, given a function \(f(x)\) and a point \(a\) at which we are interested in the limiting behavior of \(f(x)\), we want to know if there is some value \(L\) such that for any small positive number \(\epsilon\), no matter how close \(x\) is to \(a\), we can ensure that the difference \(|f(x) - L| < \epsilon\). We can now formally define the concept of a limit:
Formal Definition of Limits¶
A function \(f(x)\) has the limit \(L\) as \(x\) approaches \(a\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(x\) in the domain of \(f(x)\) with \(|x - a| < \delta\), we have \(|f(x) - L| < \epsilon\). In other words, if we can always find a small enough neighborhood around \(a\) within which the value of \(f(x)\) remains close to \(L\), then we say that the limit is \(L\).
One-Sided Limits and Limits at Infinity¶
Sometimes, we are interested in the limiting behavior of a function as \(x\) approaches \(a\) from one side only. For example, consider the function \(f(x) = |x|\). As \(x\) approaches 0 from the right, the value of \(f(x)\) is 0; however, as \(x\) approaches 0 from the left, the value of \(f(x)\) is -0, or just 0. In this case, we have two one-sided limits: \begin{align} \lim_{x \to 0^+} f(x) &= 0 \ \lim_{x \to 0^-} f(x) &= 0. \end{align} Similarly, we can define infinite and finite limits at infinity: \begin{equation} \lim_{x \to \pm \infty} f(x) = L_{\pm \infty}. \end{equation} These concepts allow us to describe the limiting behavior of a function as its input values become arbitrarily large or small.
Continuity¶
Now that we have discussed limits, let's turn our attention to continuity. A function is continuous at a point if it has no jump discontinuities, infinite discontinuities, or removable discontinuities at that point. More formally, a function \(f(x)\) is continuous at \(a\) if \begin{equation} \lim_{x \to a} f(x) = f(a). \end{equation} In other words, the value of the function at the limit point must be equal to the value of the function at the original point. If a function is continuous on an interval, it means that the function has no jump or infinite discontinuities within that interval.
Relationship Between Limits and Continuity¶
As we have seen, limits and continuity are closely related concepts in calculus. In particular, every continuous function is differentiable (i.e., its derivative exists), and every function with a finite derivative at a point is also continuous at that point. However, the converse is not true: there are functions that are differentiable but not continuous. For example, consider the function \(f(x) = |x|\). This function has a jump discontinuity at \(x=0\), so it is not continuous at that point.
Examples of Computing Limits¶
Let's now look at two examples of computing limits. The first example involves a removable discontinuity, while the second example involves a jump or infinite discontinuity:
- Compute the limit \(\lim_{x \to 2} (3x - 4)\). To do this, we can simply plug in \(x=2\) into the expression and see what happens: \begin{equation} \lim_{x \to 2} (3x - 4) = 3(2) - 4 = 0. \end{equation}
- Compute the limit \(\lim_{x \to \infty} \frac{5x^2 + x}{x}\). This limit is an example of a finite limit at infinity: \begin{align} \lim_{x \to \infty} \frac{5x^2 + x}{x} &= \lim_{x \to \infty} \left( 5x + \frac{1}{x} \right) \ &= 5x. \end{align} This limit does not exist as \(x\) approaches \(-\infty\), since the expression \(\frac{1}{x}\) becomes arbitrarily large in magnitude. However, we can see that the limit exists for positive values of \(x\).
Conclusion¶
In this article, we have explored the concepts of limits and continuity in calculus using both intuitive pictures and formal definitions. We have seen how these concepts are related to each other and how they are used to describe the limiting behavior of functions. By understanding these ideas, you will be well-equipped to tackle more advanced topics in calculus, such as derivatives, integrals, and optimization.