2.2 Limits and continuity

Limits

Definition: We say that the limit of a function $f(x)$, as $x$ approaches $a$, is $L$ if we can make the values of $f(x)$ get as close as we want to $L$ by taking $x$ sufficiently close to $a$ (but not equal to a)³. Mathematically, we can express this idea as

$$\lim_{x \to a} f(x) = L.$$

For example, if $f(x)=x^2$, then it is the case that

$$\lim_{x \to \sqrt{5}} f(x) = 5.$$

Suppose we set $x$ equal to 2.236 (this is close to $\sqrt{5}$ but not equal). Then $f(x)=$ 4.999696, which is close to 5. There is no value of $x$ other than $\sqrt{5}$ such that $f(x)=5$, but we can get as close as we want by moving $x$ closer to $\sqrt{5}$. For example, if 4.999696 isn’t close enough to satisfy us and someone demands that we be within 0.000000001 of 5, we can always accomplish that by simply moving $x$ closer to $\sqrt{5}$.

Infinite limit: A variation on this idea is to say that the limit is infinite:

$$\lim_{x \to a} f(x) = \infty.$$

This means that as $x$ gets closer to $a$, $f(x)$ keeps getting bigger, with no bound. For example, we can make $1/x^2$ be as large as we want by moving $x$ closer to 0, so $\lim_{x \to 0} 1/x^2 = \infty$ (limits of $-\infty$ are defined similarly).

One-sided limit: Sometimes, different things happen if we approach $a$ from the left or right. We say that the left-hand limit of $f(x)$ as $x$ approaches $a$ “from the left” is $L$ if $f(x)$ we can make the values of $f(x)$ as close to $L$ as we want by moving $x$ closer to $a$, but only considering points such that $x < a$. We denote this by

$$\lim_{x \to a^-} f(x) = L.$$

Right-hand limits are defined similarly. For example $\lim_{x \to 0^-} 1/x = -\infty$, whereas $\lim_{x \to 0^+} 1/x = \infty$.

The limit of $f(x)$ as $x \to a$ is $L$ if and only both the left and the right-hand limits are also $L$.

Calculating limits

Limit laws: The following laws are helpful for calculating limits. In what follows, let

$$\begin{align*} s &= \lim_{x \to a} f(x) \\ t &= \lim_{x \to a} g(x); \end{align*}$$

it is critical that these limits exist, or none of the results below necessarily hold.

$$\begin{align*} \lim_{x \to a} \{f(x) + g(x)\} &= s + t \\ \lim_{x \to a} \{f(x) - g(x)\} &= s - t \\ \lim_{x \to a} \{cf(x) + g(x)\} &= cs + t \text{ where $c$ is a constant} \\ \lim_{x \to a} \{f(x) g(x)\} &= st \\ \lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{s}{t} \text{ if } t \ne 0 \\ \lim_{x \to a} \{f(x)^n\} &= s^n \end{align*}$$

Continuity

You may have noticed that with limits, the value of $f(x)$ at $a$ is irrelevant. For example, if $f(x)=x^2$ everywhere except $x=2$, where $f(2) = -10$, it would still be the case that $\lim_{x \to 2} f(x) = 4$. In fact, $f(x)$ wouldn’t even need to be defined at 2 for this to work. If we add the requirement that $f(a)$ has to equal its limit, we end up with continuity.

Definition: A function $f$ is continuous at $a$ if

$$\lim_{x \to a} f(x) = f(a).$$

Note that this requires three things:

1. $f(a)$ is defined
2. $\lim_{x \to a} f(x)$ exists
3. These two things are equal

Expanding on this definition, we say that a function $f$ is continuous on an interval if $f$ is continuous at every number in the interval. We say that $f$ is continuous if $f$ is continuous at every point in its domain.

One-sided continuity: A function $f$ is continuous from the left at $a$ if

$$\lim_{x \to a^-} f(x) = f(a).$$

For example, consider the function

$$f(x) = \begin{cases} 0 & \text{ if } x < 0 \\ 1 & \text{ if } x \ge 0 \end{cases}$$

In this case, $f(x)$ is continuous from the right at 0, but not from the left at 0 (since $\lim_{x \to 0^-}=0$, but $f(0) = 1$).

Continuity laws: The property of continuity behaves similarly to the limit laws above. If $f(x)$ and $g(x)$ are continuous at $a$, then the following functions are also continuous at $a$:

• $f(x) + g(x)$
• $f(x) - g(x)$
• $c f(x)$, where $c$ is a constant
• $f(x) g(x)$
• $f(x) / g(x)$ if $g(a) \ne 0$

Composition: Finally, suppose that $g$ is continuous at $a$ and that $f$ is continuous at $g(a)$. Then $f(g(x))$ is continuous at $a$. In words, a continuous function of a continuous function is continuous. The function $h(x) = f(g(x))$ is known as the composition of $f$ and $g$.

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3. This section covers limits and continuity from a conceptual standpoint. For a variety of technical reasons, the definition given here isn’t actually satisfactory, and a more rigorous definition is required; see the chapter on analysis.↩︎