Limits of Sequences and Functions

A sequence is a function $a:\mathbb{N}\to\mathbb{R}$, denoted $(a_n)$ where $a_n$ is the $n$-th term. Studying convergence means asking: what happens to $a_n$ as $n$ becomes arbitrarily large?

Formal definition of convergence: $a_n \to L$ (read "$a_n$ tends to $L$") if for every $\varepsilon>0$ there exists an index $N\in\mathbb{N}$ such that for all $n>N$, $|a_n-L|<\varepsilon$. Intuitively: from some point onward, all terms stay within distance $\varepsilon$ of $L$, for any arbitrarily small $\varepsilon$.

Key examples:

• $1/n \to 0$: given $\varepsilon>0$, choose $N>1/\varepsilon$; for $n>N$, $|1/n-0|=1/n<1/N<\varepsilon$.
• $(-1)^n$ does not converge (diverges by oscillation): terms jump between $1$ and $-1$ without settling.
• $(1+1/n)^n \to e$, where $e\approx2.71828$ is Euler's number, base of natural logarithms.

Algebra of limits: if $a_n\to L$ and $b_n\to M$, then:

• $a_n+b_n \to L+M$ (limit of sum is sum of limits)
• $a_n b_n \to LM$ (limit of product is product of limits)
• $a_n/b_n \to L/M$ (if $M\neq0$) — limit of quotient is quotient of limits

Squeeze theorem (Carabinieri theorem): if $a_n\leq b_n\leq c_n$ eventually and $a_n\to L$, $c_n\to L$, then $b_n\to L$ as well. Useful for limits "sandwiched" between two convergent sequences.

Monotone convergence theorem: every monotone (increasing or decreasing) and bounded sequence converges. This follows from the completeness of $\mathbb{R}$.

$\varepsilon$-$\delta$ definition: $\lim_{x\to x_0} f(x)=L$ if for every $\varepsilon>0$ there exists $\delta>0$ such that $0<|x-x_0|<\delta \Rightarrow |f(x)-L|<\varepsilon$. Note $0<|x-x_0|$ means $x$ approaches $x_0$ but never equals it — the value of $f$ at $x_0$ itself is irrelevant to the limit.

Left and right limits: $\lim_{x\to x_0^+}f(x)$ considers $x$ approaching $x_0$ from the right ($x>x_0$); $\lim_{x\to x_0^-}f(x)$ from the left ($x<x_0$). The two-sided limit exists iff both one-sided limits exist and are equal.

Fundamental limits (to memorise):

• $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1$ — provable by the squeeze theorem: $\cos x \leq \frac{\sin x}{x} \leq 1$ for $x$ near $0$.
• $\displaystyle\lim_{x\to 0}\frac{e^x-1}{x}=1$ — follows from the definition of the derivative of the exponential.
• $\displaystyle\lim_{x\to 0}\frac{\ln(1+x)}{x}=1$ — proved by substitution $t=\ln(1+x)$.

Algebra of limits: the same rules as for sequences apply for sum, product, quotient (provided no indeterminate forms arise).

When computing a limit, direct substitution may lead to meaningless expressions: the indeterminate forms. The 7 classic forms are:

$$0/0,\quad \infty/\infty,\quad 0\cdot\infty,\quad \infty-\infty,\quad 0^0,\quad 1^\infty,\quad \infty^0.$$

Techniques for resolving indeterminate forms:

1. Algebraic manipulation: factorise, rationalise, cancel common factors. Useful for $0/0$ with polynomials.
2. Substitution using fundamental limits: recognise $\sin(t)/t$, $(e^t-1)/t$, $\ln(1+t)/t$.
3. L'Hôpital's rule (if derivatives are known): differentiate numerator and denominator separately.
4. Taylor expansions: replace functions with their approximating polynomials.

Hierarchy of growth rates as $x\to+\infty$: $\ln x \ll x^\alpha \ll e^x$ for any $\alpha>0$. This means exponential beats any power, and power beats any logarithm:

• $\displaystyle\lim_{x\to+\infty}\frac{x^n}{e^x}=0$ for every $n$
• $\displaystyle\lim_{x\to+\infty}\frac{\ln x}{x^\alpha}=0$ for every $\alpha>0$

Studying the behaviour of $f$ as $x\to\pm\infty$ or near domain boundaries is essential for sketching the graph.

Vertical asymptote at $x=x_0$: occurs when $\lim_{x\to x_0^\pm}|f(x)|=+\infty$. Typically where the denominator vanishes while the numerator does not.

Horizontal asymptote $y=L$: occurs when $\lim_{x\to\pm\infty}f(x)=L$ (finite). The curve approaches the horizontal line $y=L$ for large $|x|$.

Oblique (slant) asymptote $y=mx+q$: when no horizontal asymptote exists but the function grows linearly. Compute:

1. $m = \displaystyle\lim_{x\to\pm\infty}\frac{f(x)}{x}$ (must be finite and $\neq0$)
2. $q = \displaystyle\lim_{x\to\pm\infty}[f(x)-mx]$ (must be finite)

Note: horizontal and oblique asymptotes cannot coexist on the same side. Compute $m$ first: if $m=0$ look for horizontal, if $m\neq0$ look for oblique.