# Limits and Infinites

Limits are a concept that goes back at least as far as Archimedes, a Greek scientist, engineer and mathematician in the 3rd century BCE. He calculated a value for π based on the limit of measurements external and internal to a circle.

#### Rules for limits

If \${lim}↙{n→+∞} a_n = a\$, and \${lim}↙{n→+∞} b_n = b\$

Then, (i) \${lim}↙{n→+∞} a_n ± b_n = a ± b\$

(ii) \${lim}↙{n→+∞} a_n ⋅ b_n = a ⋅ b\$

(iii) \${lim}↙{n→+∞} ({a_n}/{b_n}) = a/b\$

If \${lim}↙{n→+∞} a_n = a\$, ⇒ \${lim}↙{n→+∞} 1/{a_n} = 1/a\$

If \${lim}↙{n→+∞} a_n = +∞\$, \${lim}↙{n→+∞} b_n = +∞\$ ⇒ \${lim}↙{n→+∞} (a_n + b_n) = +∞\$

If \${lim}↙{n→+∞} a_n = +∞\$, \${lim}↙{n→+∞} b_n = +∞\$ ⇒ \${lim}↙{n→+∞} (a_n - b_n) = ∞ - ∞\$: i.e. indeterminate solution

\${lim}↙{n→+∞} 1/{n^a} = 0\$ (a > 0)

\${lim}↙{n→+∞} a^n = 0\$ (|a| < 1)

\${lim}↙{n→+∞} ^n√{a} = 1\$ (a > 0)

\${lim}↙{n→+∞} ^n√{n} = 1\$

\${lim}↙{n→+∞} {(log n)^b}/{n^a} = 0\$ (a > 0, b ∈ ℝ)

\${lim}↙{n→+∞} {n^b}/{a^n} = 0\$ (b > 0, |a| > 1)

\${lim}↙{n→+∞} {a^n}/{n!} = 0\$ (a ∈ ℝ)

\${lim}↙{n→+∞} {n!}/{n^n} = 0\$

\${lim}↙{n→+∞} (1 + 1/n)^n = e\$

\${lim}↙{n→+∞} (1 + a/n)^{b⋅n} = e^{a⋅b}\$, (a, b ∈ ℝ)

\${lim}↙{x→±∞} (f(x) ± g(x)) = {lim}↙{x→±∞} f(x) ± {lim}↙{x→±∞} g(x) = L_1 ± L_2\$

\${lim}↙{x→±∞} (f(x) ⋅ g(x)) = {lim}↙{x→±∞} f(x) ⋅ {lim}↙{x→±∞} g(x) = L_1 ⋅ L_2\$

\${lim}↙{x→±∞} (f(x) ÷ g(x)) = {lim}↙{x→±∞} f(x) ÷ {lim}↙{x→±∞} g(x) = L_1 ÷ L_2\$, where \$L_2 ≠ 2\$

\${lim}↙{x→±∞} kf(x) = k {lim}↙{x→±∞} f(x) = kL_1\$

\${lim}↙{x→±∞} [f(x)]^{a/b} = L_1^{a/b}, a/b ∈ ℚ\$, provided \$L_1^{a/b} \$ is real

### Convergence of a Series

The sum of a finite geometric series is: \$S_n = {u_1(1-r^n)}/{1-r}\$, where \$r\$ is the common ratio of two consecutive terms, and \$n\$ is the number of terms \$u\$.

For a geometric series, \${Σ}↙{n=0}↖{∞} = {lim}↙{n→∞} {u_1(1-r^n)}/{1-r}\$.

When \$-1 < r < 1\$, \${lim}↙{n→∞} r^n = 0\$, and the series converges to its sum, \$S={u_1}/{1-r}\$.

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### Quote of the day...

Newton was not the first of the age of reason: He was the last of the magicians.