2.3 | Evaluating Limits Analytically


If \(c\) is a real number, then
\[
\lim_{x \rightarrow a} c = c
\]
\[
\lim_{x \rightarrow a} x = a
\]
Limit Laws:

  • Sum/Difference Law:
    \[
    \lim_{x \rightarrow a} \left( f(x) \pm g(x) \right) = \lim_{x \rightarrow a} f(x) \pm \lim_{x \rightarrow a} g(x)
    \]
  • Product Law:
    \[
    \lim_{x \rightarrow a} \left( f(x)g(x) \right) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x)
    \]
  • Constant Multiple Law:
    \[
    \lim_{x \rightarrow a} \left( cf(x) \right) = c\lim_{x \rightarrow a} f(x),
    \]
    for any \(c\).
  • Quotient Law:
    \[
    \lim_{x \rightarrow a} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{x \rightarrow a} f(x)}{ \lim_{x \rightarrow a} g(x)},
    \]
    provided that \(\lim_{x \rightarrow a} g(x) \not = 0\).
  • Root Law:
    \[
    \lim_{x \rightarrow a} \sqrt[n]{f(x)}= \sqrt[n]{\lim_{x \rightarrow a} f(x)},
    \]
    provided that \(n\) is a positive integer, and \(\lim_{x \rightarrow a} f(x) >0\) if \(n\) is even.
If \(n\) is a positive integer then
\[
\lim_{x \rightarrow a} \left(f(x)\right)^n = \left(\lim_{x \rightarrow a} f(x)\right)^n.
\]
\[
\lim_{x \rightarrow a} x^n = a^n
\]
If \(p(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots +a_0\) is a polynomial function, then
\[
\lim_{x \rightarrow a} p(x) = p(a)
\]
If \(f\) is a rational function defined by \(f(x) = P(x)/Q(x)\), where \(P(x)\) and \(Q(x)\) are polynomial functions and \(Q(a) \not = 0\), then
\[
\lim_{x \rightarrow a} f(x) = f(a) = \frac{P(a)}{Q(a)}
\]
Let \(a\) be a number in the domain of the given trigonometric function. Then

  • \(\displaystyle \lim_{x \rightarrow a} \sin x = \sin a \)
  • \(\displaystyle \lim_{x \rightarrow a} \cos x = \cos a \)
  • \(\displaystyle \lim_{x \rightarrow a} \tan x = \tan a \)
  • \(\displaystyle \lim_{x \rightarrow a} \csc x = \csc a \)
  • \(\displaystyle \lim_{x \rightarrow a} \sec x = \sec a \)
  • \(\displaystyle \lim_{x \rightarrow a} \cot x = \cot a \)
The Squeeze Theorem: If \(h(x) \leq f(x) \leq g(x)\) for all \(x\) in an open interval containing \(c\), except possibly at \(c\), and
\[
\lim_{x \rightarrow c} h(x) = L = \lim_{x \rightarrow c} g(x)
\]
Then
\[
\lim_{x \rightarrow c} f(x) = L
\]
Suppose that \(f(x) \leq g(x)\) for all \(x\) in an open interval containing \(a\), except possibly at \(a\), and
\[
\lim_{x \rightarrow a}f(x) = L \text{ and } \lim_{x \rightarrow a} g(x) = M
\]
Then
\[
L \leq M
\]
\[
\lim_{\theta \rightarrow 0} \frac{\sin u}{u} = 1
\]
\[
\lim_{u \rightarrow 0} \frac{\cos u -1}{u} = 0
\]
\[
\lim_{u \rightarrow 0} (1+u)^{1/u} = e
\]

E 2.3 Exercises