1.8 | Limits

We write \(\displaystyle \lim_{x \rightarrow c} f(x) = L\) if the values of \(f(x)\) approach \(L\) as \(x\) approaches \(c\).
A function \(f\) is defined on an interval around \(c\), except perhaps at the point \(x = c\). We define the limit of the function \(f(x)\) as \(x\) approaches \(c\), written \(\displaystyle \lim_{x \rightarrow c} f(x)\), to be a number \(L\) (if one exists) such that \(f(x)\) is as close to \(L\) as we want whenever \(x\) is sufficiently close to \(c\) (but \(x \not = c\). If \(L\) exists, we write
\lim_{x \rightarrow c} f(x) = L.
We define \(\displaystyle \lim_{x \rightarrow c} f(x)\) to be the number \(L\) (if one exists) such that for every \(\epsilon >0\) (as small as we want), there is a \(\delta >0\) (sufficiently small) such that \(|x-c| < \delta\) and \(x \not = c\), then \(|f(x) - L| < \epsilon\).
Properties of Limits:
Assuming all the limits on the right-hand side exists:

  1. If \(b\) is a constant, then \(\displaystyle \lim_{x \rightarrow c} \left( b f(x) \right) = b \left( \lim_{x \rightarrow c} f(x) \right) \).
  2. \(\displaystyle \lim_{x \rightarrow c} \left( f(x) + g(x) \right) = \lim_{x \rightarrow c} f(x)+\lim_{x \rightarrow c} g(x)\).
  3. \(\displaystyle \lim_{x \rightarrow c} \left( f(x) g(x) \right) = \left( \lim_{x \rightarrow c} f(x) \right) \left(\lim_{x \rightarrow c} g(x) \right)\).
  4. \(\displaystyle \lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)}\), provided \(\displaystyle \lim_{x \rightarrow c} g(x) \not = 0\).
  5. For any constant \(k\), \(\displaystyle \lim_{x \rightarrow c} k = k\).
  6. \( \displaystyle \lim_{x \rightarrow c} x = c\).
If \(f(x)\) gets as close to a number \(L\) as we please when \(x\) gets sufficiently large, then we write
\lim_{x \rightarrow \infty} f(x) = L.
Similarly, if \(f(x)\) approaches \(L\) when \(x\) is negative and has a sufficiently large absolute value, then we write
\lim_{x \rightarrow – \infty} f(x) = L.
The function \(f\) is continuous at \(x = c\) if \(f\) is defined at \(x = c\) and if
\lim_{x \rightarrow c} f(x) = f(c).
In other words, \(f(x)\) is as close as we want to \(f(c)\) provided \(x\) is close enough to \(c\). The function is continuous on the interval \([a,b]\) if it is continuous at every point of the interval. If \(c\) is an endpoint of the interval, we define continuity at \(x=c\) using one-sided limits at \(c\).
Suppose that \(f\) and \(g\) are continuous on an interval and the \(b\) is a constant. Then, on that same interval.

  1. \(bf(x)\) is continuous.
  2. \(f(x) + g(x)\) is continuous.
  3. \(f(x)g(x)\) is continuous.
  4. \(f(x)/g(x)\) is continuous, provided \(g(x) \not = 0\) on the interval.
If \(f\) and \(g\) are continuous, and if the composite function \(f(g(x))\) is defined on an interval, then \(f(g(x))\) is continuous on that interval.

E 1.8 Exercises