maths > calculus-limits

Limits involving Binomial Expressions

what you'll learn...

Limits involving Binomial Expressions

»  Special case of canceling factors in numerator and denominator
→  limxaxnanxa=nan1${\lim}_{x \to a} \frac{{x}^{n} - {a}^{n}}{x - a} = n {a}^{n - 1}$

»  With change of variable x=1+y$x = 1 + y$ and constant a=1$a = 1$
→  limy0(1+y)n1y=n${\lim}_{y \to 0} \frac{{\left(1 + y\right)}^{n} - 1}{y} = n$

recap

Factoring binomials refers to factoring of anbn${a}^{n} - {b}^{n}$.

The word binomial means: of two terms.

The expression anbn${a}^{n} - {b}^{n}$ is a two variate binomial of degree $n$.
two variate : $a$ and $b$ are two variables
binomial : ${a}^{n}$ and $- {b}^{n}$ are the two terms
degree $n$: The maximum power is $n$

Factoring the binomials is given by
${a}^{n} - {b}^{n}$

example

The value of $f \left(x\right) = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{4} - 16}}{\textcolor{c \mathmr{and} a l}{x - 2}}$ at $x = 2$ is $\frac{0}{0}$ by direct substitution $x = 2$.

The limit of $f \left(x\right) = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{4} - 16}}{\textcolor{c \mathmr{and} a l}{x - 2}}$ at $x = 2$ is $4 \times {2}^{3}$.

limit of $f \left(x\right) = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{4} - 16}}{\textcolor{c \mathmr{and} a l}{x - 2}}$ at $x = 2$.
On substitution of $x = 2$ the function evaluates to $\frac{0}{0}$

Limit of the function is
${\lim}_{x \to 2} \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{4} - 16}}{\textcolor{c \mathmr{and} a l}{x - 2}}$
$\quad \quad = {\lim}_{x \to 2} \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{4} - {2}^{4}}}{\textcolor{c \mathmr{and} a l}{x - 2}}$
$\quad \quad = {\lim}_{x \to 2} \frac{\textcolor{\mathrm{de} e p s k y b l u e}{\left(x - 2\right) \left({x}^{3} + 2 {x}^{2} + 4 x + 8\right)}}{\textcolor{c \mathmr{and} a l}{x - 2}}$
$\quad \quad = {\lim}_{x \to 2} \left({x}^{3} + 2 {x}^{2} + 4 x + 8\right)$
$\quad \quad = {2}^{3} + 2 \times {2}^{2} + 4 \times 2 + 8$
$\quad \quad = 4 \times {2}^{3}$

note: $\left(x - 2\right)$ is not canceled at $x = 2$. But, limit is applied to $\frac{x - 2}{x - 2}$

To find limit of functions with binomials, factor the binomials to cancel out the factor involving $0$.

What is the limit for $f \left(x\right) = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{15} - 1}}{\textcolor{c \mathmr{and} a l}{{x}^{10} - 1}}$ at $x = 1$
The answer is '$\frac{3}{2}$'

summary

Limit of expressions with Binomials: For any positive integer n
${\lim}_{x \to a} \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{n} - {a}^{n}}}{\textcolor{c \mathmr{and} a l}{x - a}} = n {a}^{n - 1}$

Outline