Limits Notes | crackin

LimitsDefinition of limits

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Solve it on paper, then

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Solution continues

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

Solve it on paper, then

Solution continues

Solve it on paper, then

After the notes

Formula sheet

23 results from this chapter, grouped the way the notes teach them.

Left hand limit at $x=a$: $\displaystyle\lim_{x\to a^-}f(x)=\lim_{h\to0}f(a-h)$
Right hand limit at $x=a$: $\displaystyle\lim_{x\to a^+}f(x)=\lim_{h\to0}f(a+h)$
When does $\displaystyle\lim_{x\to a}f(x)$ exist?: L.H.L. $=$ R.H.L. $=$ a finite value — whether or not $f(a)$ is defined
Deleted neighbourhood of $a$: $(a-\delta,\,a+\delta)-\{a\}$ , $\delta$ a small positive number
One side of $a$ outside the domain: The limit is the limit from the other side, e.g. $\displaystyle\lim_{x\to0}\sqrt{x}=0$

The 7 indeterminate forms: $\frac00,\ \frac{\infty}{\infty},\ 0\times\infty,\ \infty-\infty,\ 1^{\infty},\ 0^0,\ \infty^0$
$a\pm\infty\ \ (a\in\mathbb{R})$: $\pm\infty$
$\infty+\infty$ and $-\infty-\infty$: $+\infty$ and $-\infty$
$a\times(+\infty)\ \ (a\neq0)$: $+\infty$ if $a>0$ · $-\infty$ if $a<0$
$\dfrac{a}{\pm\infty}$: $0$
$\dfrac{a}{0}\ \ (a\neq0)$: $+\infty$ if $a>0$ · $-\infty$ if $a<0$ — check both sides of $0$
$0^{\infty}$: $0$

$\displaystyle\lim_{x\to a}\frac{x^n-a^n}{x-a}$: $na^{n-1}$ (for every rational $n$ )
$\dfrac{x^n-a^n}{x-a}\ \ (n\in\mathbb{N})$ expands to: $x^{n-1}+x^{n-2}a+\dots+xa^{n-2}+a^{n-1}$
$\displaystyle\lim_{x\to1}\frac{x^m-1}{x^n-1}$: $\dfrac{m}{n}$
$\displaystyle\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}$: $\dfrac12$ (rationalise the numerator)
$\displaystyle\lim_{x\to0}x\sin\frac1x$: $0$ — zero times a bounded quantity
$\displaystyle\lim_{n\to\infty}\frac{1^2+2^2+\dots+n^2}{n^3}$: $\dfrac13$

Method: Divide numerator and denominator by the highest power of $x$ — $\frac{c}{x^k}\to0$
Degree of numerator $=$ degree of denominator: Ratio of the leading coefficients
Degree of numerator $<$ degree of denominator: $0$
Degree of numerator $>$ degree of denominator: $\pm\infty$ — the limit does not exist
$\sqrt{x^2}$ as $x\to-\infty$: $|x|=-x$ — keep the sign when taking $x$ out of a root

Last look

The checks to run before you sit a question on functions.

1Always substitute first; only an indeterminate form needs a method.
2 $\frac00$ : factorise and cancel the factor that causes the zero, or rationalise when there is a root.
3Roots in both numerator and denominator → rationalise both (double rationalisation).
4 $\infty-\infty$ with roots → multiply and divide by the conjugate to get $\frac{\infty}{\infty}$ .
5For $\lim_{x\to\infty}$ of a polynomial ratio, only the highest-degree terms matter.
6If the denominator $\to0$ and the limit is finite, the numerator must also $\to0$ — use it to find unknown constants.
7Greatest integer, $\{x\}$ , $\operatorname{sgn}$ , $|x|$ and $\tan$ near $\frac{\pi}{2}$ : check L.H.L. and R.H.L. separately.
8At a point where $f(x)\to\pm\infty$ from the two sides (e.g. $\frac{2}{x}$ at $0$ ), the limit does not exist.
9A substitution such as $x=\frac1y$ or $\cot x=t$ can turn an awkward limit into a standard one.
10L'Hospital's rule ( $\lim\frac fg=\lim\frac{f'}{g'}$ ) applies only to $\frac00$ and $\frac{\infty}{\infty}$ .