# 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.