Limit

Mort Yao

1 \((ε, δ)\)-definition

Definition 1.1. Let \(f : D \to \mathbb{R}\) be a function defined on a subset \(D \subseteq \mathbb{R}\), let \(a\) be a limit point of \(D\) and \(L \in \mathbb{R}\). We say the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\), denoted \[\lim_{x \to a} f(x) = L\] if for all \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \[|f(x) - L| < \varepsilon\] whenever \[0 < |x - a| < \delta\]

Definition 1.2. (One-sided limits) Let \(f : D \to \mathbb{R}\) be a function defined on a subset \(D \subseteq \mathbb{R}\), let \(a\) be a limit point of \(D\) and \(L \in \mathbb{R}\). We say the left-hand limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\), denoted \[\lim_{x \to a^-} f(x) = L\] if for all \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \[|f(x) - L| < \varepsilon\] whenever \[a - \delta < x < a\]

We say the right-hand limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\), denoted \[\lim_{x \to a^+} f(x) = L\] if for all \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \[|f(x) - L| < \varepsilon\] whenever \[a < x < a + \delta\]

Definition 1.3. (Infinite limits) Let \(f : D \to \mathbb{R}\) be a function defined on a subset \(D \subseteq \mathbb{R}\), let \(a\) be a limit point of \(D\). We say the limit of \(f(x)\) as \(x\) approaches \(a\) is infinity, denoted \[\lim_{x \to a} f(x) = \infty\] if for all \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \[f(x) > \varepsilon\] whenever \[0 < |x - a| < \delta\]

We say the limit of \(f(x)\) as \(x\) approaches \(a\) is negative infinity, denoted \[\lim_{x \to a} f(x) = -\infty\] if for all \(\varepsilon > 0\) there exists a \(\delta > 0\) such that \[f(x) < -\varepsilon\] whenever \[0 < |x - a| < \delta\]

Definition 1.4. (Limits at infinity) Let \(f : D \to \mathbb{R}\) be a function defined on a subset \(D \subseteq \mathbb{R}\), let \(L \in \mathbb{R}\). We say the limit of \(f(x)\) as \(x\) approaches infinity is \(L\), denoted \[\lim_{x \to \infty} f(x) = L\] if for all \(\varepsilon > 0\) there exists a \(c\) such that \[|f(x) - L| < \varepsilon\] whenever \[x > c\]

We say the limit of \(f(x)\) as \(x\) approaches negative infinity is \(L\), denoted \[\lim_{x \to -\infty} f(x) = L\] if for all \(\varepsilon > 0\) there exists a \(c\) such that \[|f(x) - L| < \varepsilon\] whenever \[x < c\]

Limits and asymptotic analysis. Limits involving infinity are often connected with the concept of asymptotes.

The above definitions can be combined in a natural way to produce definitions for different combinations, such as:

Example 1.5. \(\lim_{x \to \infty} x = \infty\).

Example 1.6. \(\lim_{x \to 0^+} \ln x = -\infty\).

2 Limit Rules

Theorem 2.1. (Constant rule) If \(a\), \(c\) are constants then \(\lim_{x \to a} c = c\).

Theorem 2.2. (Identity rule) If \(a\) is a constant then \(\lim_{x \to a} x = a\).

Theorem 2.3. (Scalar product rule) Suppose that \(\lim_{x \to a} f(x) = L\) for finite \(L\) and that \(k\) is a constant, then \(\lim_{x \to a} k \cdot f(x) = k \cdot \lim_{x \to a} f(x) = k \cdot L\).

Theorem 2.4. (Algebraic limit theorem) \[\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\] \[\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)\] \[\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\] \[\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\]

Theorem 2.5. (Squeeze theorem) If \(g(x) \leq f(x) \leq h(x)\) holds for all \(x\) in an open interval containing \(a\), except possibly at \(x=a\) itself, and \(\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L\), then \[\lim_{x \to a} f(x) = L\]

Theorem 2.6. (Composition rule) If \(f(x)\) is continuous at \(x=b\) and \(\lim_{x \to a} g(x) = b\), then \[\lim_{x \to a} f(g(x)) = f(b) = f\big(\lim_{x \to a} g(x)\big)\]

*Theorem 2.7. (L'Hôpital's rule) Let \(f(x)\) and \(g(x)\) be differentiable functions defined on an open interval except possibly at \(x=a\) itself, if:

  1. \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0 \textrm{ or } \pm\infty\)
  2. \(g'(x) \neq 0\) for all \(x \neq a\)
  3. \(\lim_{x \to a} \frac{f'(x)}{g'(x)} = L\)

Then \[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} = L\]

*Example 2.8. \(\lim_{x \to 1} \frac{\ln x}{x - 1} = \lim_{x \to 1} \frac{(\ln x)'}{(x - 1)'} = \lim_{x \to 1} \frac{1/x}{1} = 1\).

*Example 2.9. \(\lim_{x \to \infty} \frac{e^x}{x} = \lim_{x \to \infty} \frac{(e^x)'}{x'} = \lim_{x \to \infty} \frac{e^x}{1} = \infty\).

3 Computation of Limits (Using Maxima)

Example 3.1. Compute \(\lim_{x \to 1} \frac{\ln x}{x - 1}\):

 limit(log(x) / (x - 1), x, 1);

Example 3.2. Compute \(\lim_{x \to \infty} \frac{e^x}{x}\):

 limit(exp(x) / x, x, inf);

Example 3.3. Compute one-sided limits \(\lim_{x \to \frac{\pi}{2}^+} \tan x\) and \(\lim_{x \to \frac{\pi}{2}^-} \tan x\):

 limit(tan(x), x, %pi / 2, plus);
 limit(tan(x), x, %pi / 2, minus);

4 Limits of Special Interest

Theorem 4.1. (Asymptotes for rational functions) Suppose that all \(a_i\) and \(b_i\) are constants, \(n \in \mathbb{N}\), then \[\lim_{x \to \infty} \frac{a_1x^n + a_2x^{n-1} + \dots + a_n}{b_1x^n + b_2x^{n-1} + \dots + b_n} = \frac{a_1}{b_1}\]

Theorem 4.2. (Euler's number) \[\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e\]

where \(e = \sum_{n=0}^\infty \frac{1}{n!} \approx 2.718 \dots\) is a mathematical constant. See also some inequalities related to Euler's number.