Approximating tangents

Motivation

Find the slope of

f (x) = \frac{x^{2}}{2}

at $P = [1, \frac{1}{2}]$ . Find the equation of the line touching $f (x)$ at $P$ .

Secant

The secant to $f (x)$ at the points $[ω, f (ω)]$ and $[u, f (u)]$ is the line passing through these points. Its slope is

𝓂 = \frac{f (ω) - f (u)}{ω - u}

lim_{x \to 1} \frac{f (1) - f (x)}{1 - x} = lim_{x \to 1} \frac{\frac{1^{2}}{2} - \frac{x^{2}}{2}}{1 - x} = lim_{x \to 1} \frac{(1 - x) (1 + x)}{2 (1 - x)}

interactive solution

Limit

Let $f : 𝒟 \to ℛ$ be a function such that $f (x)$ is defined on a neighborhood of $ω$ (except possibly at $ω$ itself). Then $ℒ$ is a limit for $f$ at $ω$

\lim_{x \to ω} f (x) = ℒ

\forall ε \in ℝ^{+}, \exists δ (ε) \in ℝ^{+} : 0 < d (x, ω) < δ (ε) \Rightarrow d (f (x), ℒ) < ε

Real valued functions

Let $f : ℝ \to ℝ$ be defined on $(ω - ℎ, ω) \cup (ω, ω + ℎ)$ . Then $ℒ$ is a limit for $f$ at $ω$

\lim_{x \to ω} f (x) = ℒ

\forall ε \in ℝ^{+}, \exists δ (ε) \in ℝ^{+} : 0 < | x - ω | < δ (ε) \Rightarrow | f (x) - ℒ | < ε

Right Limit

Let $f : ℝ \to ℝ$ be defined on $(ω, ω + ℎ)$ . Then $ℒ^{+}$ is a right limit (limit from the right) for $f$ at $ω$

\lim_{x \to ω^{+}} f (x) = ℒ^{+}

\forall ε \in ℝ^{+}, \exists δ (ε) \in ℝ^{+} : 0 < x - ω < δ (ε) \Rightarrow | f (x) - ℒ^{+} | < ε

Left Limit

Let $f : ℝ \to ℝ$ be defined on $(ω - ℎ, ω)$ . Then $ℒ^{-}$ is a left limit (limit from the left) for $f$ at $ω$

\lim_{x \to ω^{-}} f (x) = ℒ^{-}

\forall ε \in ℝ^{+}, \exists δ (ε) \in ℝ^{+} : 0 < ω - x < δ (ε) \Rightarrow | f (x) - ℒ^{-} | < ε

\lim_{x \to ω} f (x) = ℒ \Leftrightarrow \lim_{x \to ω^{-}} f (x) = ℒ^{-} \land \lim_{x \to ω^{+}} f (x) = ℒ^{+} \land ℒ^{-} = ℒ^{+}

argument

Positive Infinity Limit

Let $f : ℝ \to ℝ$ be defined on $(ω - ℎ, ω) \cup (ω, ω + ℎ)$ . Then $\infty$ is a limit for $f$ at $ω$

\lim_{x \to ω} f (x) = \infty

\forall ℬ \in ℝ^{+}, \exists δ (ℬ) \in ℝ^{+} : 0 < | x - ω | < δ (ℬ) \Rightarrow ℬ < f (x)

Right Positive Infinity Limit

Let $f : ℝ \to ℝ$ be defined on $(ω, ω + ℎ)$ . Then $\infty$ is a right limit for $f$ at $ω$

\lim_{x \to ω^{+}} f (x) = \infty

\forall ℬ \in ℝ^{+}, \exists δ (ℬ) \in ℝ^{+} : 0 < x - ω < δ (ℬ) \Rightarrow ℬ < f (x)

Left Positive Infinity Limit

Let $f : ℝ \to ℝ$ be defined on $(ω - ℎ, ω)$ . Then $\infty$ is a left limit for $f$ at $ω$ .

\lim_{x \to ω^{-}} f (x) = \infty

\forall ℬ \in ℝ^{+}, \exists δ (ℬ) \in ℝ^{+} : 0 < ω - x < δ (ℬ) \Rightarrow ℬ < f (x)

Negative Infinity Limit

Let $f : ℝ \to ℝ$ be defined on $(ω - ℎ, ω) \cup (ω, ω + ℎ)$ . Then $- \infty$ is a limit for $f$ at $ω$

\lim_{x \to ω} f (x) = - \infty

\forall ℬ \in ℝ^{+}, \exists δ (ℬ) \in ℝ^{+} : 0 < | x - ω | < δ (ℬ) \Rightarrow f (x) < - ℬ

Right Negative Infinity Limit

Let $f : ℝ \to ℝ$ be defined on $(ω, ω + ℎ)$ . Then $- \infty$ is a right limit for $f$ at $ω$

\lim_{x \to ω^{+}} f (x) = - \infty

\forall ℬ \in ℝ^{+}, \exists δ (ℬ) \in ℝ^{+} : 0 < x - ω < δ (ℬ) \Rightarrow f (x) < - ℬ

Left Negative Infinity Limit

Let $f : ℝ \to ℝ$ be defined on $(ω - ℎ, ω)$ . Then $- \infty$ is a left limit for $f$ at $ω$

\lim_{x \to ω^{-}} f (x) = - \infty

\forall ℬ \in ℝ^{+}, \exists δ (ℬ) \in ℝ^{+} : 0 < ω - x < δ (ℬ) \Rightarrow f (x) < - ℬ

Vertical asymptotes

Let $f : ℝ \to ℝ$ then $x = ω$ is a vertical asymptote for $f (x)$ if one of the following holds:

\begin{matrix} \lim_{x \to ω} f (x) = \infty & \lim_{x \to ω} f (x) = - \infty \\ \lim_{x \to ω^{+}} f (x) = \infty & \lim_{x \to ω^{+}} f (x) = - \infty \\ \lim_{x \to ω^{-}} f (x) = \infty & \lim_{x \to ω^{-}} f (x) = - \infty \end{matrix}

Limit at Positive Infinity

Let $f : ℝ \to ℝ$ be defined on $(ℎ, \infty)$ . Then $ℒ$ is a limit for $f$ at $\infty$

\lim_{x \to \infty} f (x) = ℒ

\forall ε \in ℝ^{+}, \exists ℳ (ε) \in ℝ^{+} : ℳ (ε) < x \Rightarrow | f (x) - ℒ | < ε

Infinite limits at positive infinity

\lim_{x \to \infty} f (x) = \infty

\forall ℬ \in ℝ^{+}, \exists ℳ (ℬ) \in ℝ^{+} : ℳ (ℬ) < x \Rightarrow ℬ < f (x)

\lim_{x \to \infty} f (x) = - \infty

\forall ℬ \in ℝ^{+}, \exists ℳ (ℬ) \in ℝ^{+} : ℳ (ℬ) < x \Rightarrow f (x) < - ℬ

Limit at Negative Infinity

Let $f : ℝ \to ℝ$ be defined on $(- \infty, ℎ)$ . Then $ℒ$ is a limit for $f$ at $\infty$

\lim_{x \to - \infty} f (x) = ℒ

\forall ε \in ℝ^{+}, \exists ℳ (ε) \in ℝ^{+} : x < - ℳ (ε) \Rightarrow | f (x) - ℒ | < ε

Infinite limits at negative infinity

\lim_{x \to - \infty} f (x) = \infty

\forall ℬ \in ℝ^{+}, \exists ℳ (ℬ) \in ℝ^{+} : x < - ℳ (ℬ) \Rightarrow ℬ < f (x)

\lim_{x \to - \infty} f (x) = - \infty

\forall ℬ \in ℝ^{+}, \exists ℳ (ℬ) \in ℝ^{+} : x < - ℳ (ℬ) \Rightarrow f (x) < - ℬ

Horizontal asymptotes

Let $f : ℝ \to ℝ$ and $ℒ \in ℝ$ then $y = ℒ$ is a horizontal asymptote for $f (x)$ if one of the following holds:

\begin{matrix} \lim_{x \to \infty} f (x) = ℒ \\ \lim_{x \to - \infty} f (x) = ℒ \end{matrix}

Examples

Limit examples

try $lim_{x \to 3} (2 + \frac{3}{x})$
try $lim_{x \to 3} (1 + ℯ^{\frac{3 - x}{2}})$
try $lim_{x \to 2} (3 - \frac{x}{2} + \frac{| x - 2 |}{2 (x - 2)})$
try $lim_{x \to - 2} \frac{1}{{(x + 2)}^{2}}$
try $lim_{x \to 0} sin (\frac{π}{x})$

One sided limits

try $lim_{x \to 1^{+}} f (x)$ where $f (x) = {\begin{matrix} \frac{1}{1 - x} & x < 1 \\ \frac{1 + x}{x} & 1 < x \end{matrix}$
try $lim_{x \to 2^{-}} f (x)$ where $f (x) = {\begin{matrix} 2 + \frac{1}{x - 3} & x < 2 \\ 4 - \frac{x}{2} & 2 \leq x \end{matrix}$
try $lim_{x \to 0^{-}} f (x)$ and $lim_{x \to 0^{+}} f (x)$ where
$f (x) = {\begin{matrix} sin \frac{4 π}{x} & x < 0 \\ \sqrt{x^{3}} & 0 < x \end{matrix}$

$+∞$ limits

try $lim_{x \to - 2} \frac{1}{{(x - 2)}^{2}}$
try $lim_{x \to 0^{+}} f (x)$ where $f (x) = {\begin{matrix} 2 + \frac{1}{x^{2} + 1} & x \leq 0 \\ \frac{1}{x} & 0 < x \end{matrix}$
try $lim_{x \to 0^{-}} f (x)$ where $f (x) = \frac{-1}{x}$

$-∞$ limits

try $lim_{x \to 2} \frac{-1}{| x + 2 |}$
try $lim_{x \to 2^{+}} \frac{-1}{| x + 2 |}$
try $lim_{x \to 2^{-}} f (x)$ where $f (x) = {\begin{matrix} \frac{1}{x - 2} & x < 2 \\ \frac{x}{3} & 2 < x \end{matrix}$

Limits at $+∞$

try $lim_{x \to \infty} (2 sin x + 6)$
and $lim_{x \to \infty} f (x)$ where $f (x) = {\begin{matrix} sin x & x \leq 2 \\ \frac{5 cos x}{x} + 2 & 2 < x \end{matrix}$
try $lim_{x \to \infty} f (x) = \frac{x}{10} (7 + sin x)$

Infinite Limits at $+∞$

try $lim_{x \to \infty} \frac{x}{10} (7 + sin x)$
try $lim_{x \to \infty} \frac{x}{2} (1 + sin x)$
try $lim_{x \to \infty} -4 \sqrt{x}$

Limits at $-∞$

try $lim_{x \to -∞} f (x)$ where $f (x) = {\begin{matrix} \frac{1}{x} + 3 & x < -1 \\ \frac{2}{1 + x^{2}} + 1 & -1 < x \end{matrix}$
try $lim_{x \to -∞} sin x$

Infinite Limits at $-∞$

try $lim_{x \to -∞} 8 log (- x)$
try $lim_{x \to -∞} -8 log (- x)$