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:
- \(f(a)\) is defined
- \(\lim_{x \to a} f(x)\) exists
- 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 \(g\).