Lesson 0011: Limits of Special Functions

Theorem: Spec. monom. For $x>1$ we have $\frac{1}{k^x}\to 0$.
Statement and proof. Statement: Let $x\in\Bbb Q$ and $x>1$.
For $\varepsilon\in\Bbb R^+$ let $\left(\frac{1}{k}\right)_{k=p}$ be within $\varepsilon^{1/x}$ of 0.
Then for $k\ge p$ we have $$ \frac{1}{k} < \varepsilon^{1/x} $$ therefore $$ \frac{1}{k^x} < \varepsilon $$ which shows $\frac{1}{k^x} \to 0$.
$\Box$
Theorem: Infin. lim. poly.
The limit of every nonconstant polynomial is infinite.
Statement and proof. Statement: Let $P(k)$ be a polynomial with degree $n\ge 1$, with a positive leading coefficient. Then $\lim_{k\to\infty}P(k) = \infty$. If the leading coefficient is negative then $\lim_{k\to\infty} P(k) = -\infty$ and otherwise $\lim_{k\to\infty}P(k)=\infty$.

Proof: Let $P(k) = a_0+a_1k + \cdots + a_nk^n = k^n\cdot (a_0/k^n + a_1/k^{n-1} + \cdots + a_n)$.
Then $$\begin{aligned} \lim_{k\to\infty} P(k) = \infty\cdot a_n \end{aligned}$$ If $a_n < 0$ then $\infty \cdot a_n = -\infty$ and otherwise $\infty\cdot a_n = \infty$.
$\Box$
Theorem: Spec. rational func.
The limit of rationals functions.
Statement and proof. Statement: Suppose $P(k) = a_0+a_1k+\cdots+a_{m-1}k^{m-1}+k^m$ is a monic polynomial of degree $m$.
And suppose $Q(k) = b_0+b_1k+\cdots+b_{n-1}k^{n-1}+k^n$ is a monic polynomial of degree $n$. Also assume that $Q(k)\ne 0$ for any $k\ge m$.
Let $R = \frac{P}{Q}$. Proof:
If $m = n$ then $\lim_{k\to\infty} R(k) = 1$. After factoring $k^m$ from the numerator and denominator, we have $$ R(k) = \frac{a_0/k^m+a_1/k^{m-1}+\cdots+a_{m-1}/k + 1}{b_0/k^m + b_1/k^{m-1}+\cdots+b_{m-1}/k+1} $$ Applying the quotient rule and the fact that $\lim_{k\to\infty}\frac{1}{k^q} = 0$ for every integer $q \ge 1$, we obtain $$\lim_{k\to\infty} R(k) = \frac{0+0+\cdots+0+1}{0+0+\cdots + 0+1} $$
If $m < n$ then $\lim_{k\to\infty} R(k) = 0$. After factoring, $$ R(k) = \left(\frac{1}{k^{n-m}}\right)\cdot \left( \frac{a_0/k^m + a_1/k^{m-1}+\cdots+a_{m-1}/k + 1}{b_0/k^n+b_1/k^{n-1}+\cdots+b_{n-1}/k + 1} \right) $$ Then $\lim_{k\to\infty} R(k) = 0\cdot 1$.
If $m > n$ then $\lim_{k\to\infty}R(k) = \infty$. After factoring, $$ R(k) = ( n^{m-n} ) \cdot \left( \frac{a_0/k^m + a_1/k^{m-1} + \cdots+ a_{m-1}/k+1}{b_0/k^n + b_1/k^{n-1} + \cdots + b_{n-1}/k + 1} \right) $$ Then $\lim_{k\to\infty} R(k) = \infty \cdot 1 = \infty$.
$\Box$
Theorem: Spec. exp.
Limits of exponentials.
Statement and proof. Statement: Let $0 < c < 1$, then $\lim_{k\to\infty} c^k = 0$. If $1 < c$ then $\lim_{k\to\infty} c^k = \infty$.

Proof: Let $\varepsilon\in\Bbb R^+$ and let $(c^k)_{k=1}