 maths > calculus-limits

Algebra of Limits

what you'll learn...

Understanding Algebra of Limits

»  Finding limit of function as sub-expressions

→  f(x)±g(x)$f \left(x\right) \pm g \left(x\right)$

→  f(x)×g(x)$f \left(x\right) \times g \left(x\right)$

→  f(x)÷g(x)$f \left(x\right) \div g \left(x\right)$

→  [f(x)]n${\left[f \left(x\right)\right]}^{n}$

→  f(x)$f \left(x\right)$ and y=g(x)$y = g \left(x\right)$

algebra is about operations + - * / ^

"Algebra of limits" means: Properties to find limit of functions given as algebraic operations of several functions.

Let us see this in detail.

The basic mathematical operations are
•  multiplication and division
•  powers, roots, and logarithms.

Two or more function $g \left(x\right)$ $h \left(x\right)$ can form another function $f \left(x\right)$.
$f \left(x\right) = g \left(x\right) \star h \left(x\right)$ $\quad \quad$ where $\star$ is one of the mathematical operations.

Will there be any relationship between the limits of the functions $\lim g \left(x\right)$ ; $\lim h \left(x\right)$ and the limit of the function $\lim f \left(x\right)$?
Algebra of limits analyses this and provides the required knowledge.

caution before using

In computing limit of a function, the value of the function or limit of the function changes :

•  when a function evaluates to $0$ in denominator

•  When a function evaluates to $\infty$

•  at the discontinuous points of piecewise functions.

When applying algebra of limits to elements of a function, look out for the following cases.

•  Expressions evaluating to $\frac{1}{0}$ or $\frac{0}{0}$ or $\infty \times 0$ or $\frac{\infty}{\infty}$

eg: $\frac{1}{x - 1}$, $\frac{{x}^{2} - 1}{x - 1}$, $\tan x \cot x$, $\frac{\tan x}{\sec x}$

•  Expressions evaluating to $\infty - \infty$ or $\infty + \left(- \infty\right)$
eg: $\frac{{x}^{2} - 4 x}{x} - \frac{1}{x}$
•  discontinuous points of piecewise functions

eg: $\left\{\begin{matrix}1 & \quad \mathmr{if} \quad x > 0 \\ 0 & \quad \mathmr{if} \quad x \le 0\end{matrix}\right.$

The algebra of limit applies only when the above values do not occur.

Example:
${\lim}_{x \to 1} \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{2} - 1}}{\textcolor{c \mathmr{and} a l}{x - 1}}$
$\quad \quad \ne \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{\lim}_{x \to 1} \left({x}^{2} - 1\right)}}{\textcolor{c \mathmr{and} a l}{{\lim}_{x \to 1} \left(x - 1\right)}}$
The above is not applicable because it evaluates to $\frac{0}{0}$.

Algebra of limits helps to simplify finding limit by applying the limit to sub-expressions of a function.
Algebra of limits may not be applicable to the sub-expressions evaluating to $0$ or $\infty$ or at discontinuities.

summary

Algebra of Limits: If a function $f \left(x\right)$ consists of mathematical operations of sub-expressions ${f}_{1} \left(x\right)$, ${f}_{2} \left(x\right)$, etc. then the limit of the function can be applied to the sub-expressions.

If any of the sub-expressions or combination of them evaluate to $0$ or $\infty$ then, the algebra of limit may not be applied to those sub-expressions.

results

limit of sum (or difference) is sum (or difference) of limits.

Limit of Sum or Difference: Given that ${\lim}_{x \to a} f \left(x\right)$ and ${\lim}_{x \to a} g \left(x\right)$ exists. Then
${\lim}_{x \to a} \left(f \left(x\right) \pm g \left(x\right)\right)$
$\quad \quad = {\lim}_{x \to a} f \left(x\right) \pm {\lim}_{x \to a} g \left(x\right)$

limit of product is product of limits.

Limit of Product: Given that ${\lim}_{x \to a} f \left(x\right)$ and ${\lim}_{x \to a} g \left(x\right)$ exists. Then
${\lim}_{x \to a} \left(f \left(x\right) \cdot g \left(x\right)\right)$
$\quad \quad = {\lim}_{x \to a} f \left(x\right) \cdot {\lim}_{x \to a} g \left(x\right)$

limit of quotient is quotient of limits.

Limit of Quotient: Given that ${\lim}_{x \to a} f \left(x\right)$ and ${\lim}_{x \to a} g \left(x\right)$ exists. Then
${\lim}_{x \to a} \left(\frac{f \left(x\right)}{g \left(x\right)}\right)$
$\quad \quad = \frac{{\lim}_{x \to a} f \left(x\right)}{{\lim}_{x \to a} g \left(x\right)}$

limit of exponent is exponent of limit.

Limit of Exponent: Given that ${\lim}_{x \to a} f \left(x\right)$ exists. Then
${\lim}_{x \to a} \left[f {\left(x\right)}^{n}\right]$
$\quad \quad = {\left[{\lim}_{x \to a} f \left(x\right)\right]}^{n}$

limit of root is root of limit.

Limit of Root: Given that ${\lim}_{x \to a} f \left(x\right)$ and ${\lim}_{x \to a} g \left(x\right)$ exists. Then
${\lim}_{x \to a} \left[f {\left(x\right)}^{\frac{1}{n}}\right]$
$\quad \quad = {\left[{\lim}_{x \to a} f \left(x\right)\right]}^{\frac{1}{n}}$

The variable in a limit can be changed.

Given
${\lim}_{x \to 0} \frac{\sin x}{x} = 1$ ;
${\lim}_{x \to 0} \frac{\sin \left({x}^{2}\right)}{x}$
$\quad \quad = {\lim}_{x \to 0} x \frac{\sin \left({x}^{2}\right)}{{x}^{2}}$
$\quad \quad = {\lim}_{x \to 0} x \times {\lim}_{y \to 0} \frac{\sin y}{y}$
where $y = {x}^{2}$
by that definition, ${\lim}_{x \to 0}$ changes to ${\lim}_{y \to 0}$ . $\quad \quad = 0 \times 1$
$\quad \quad = 0$

Note: If, in another case, $y = \cos x$ then ${\lim}_{x \to 0}$ changes to ${\lim}_{y \to 1}$, as $y = \cos 0 = 1$.

Change of variable in a Limit: Given that $y = g \left(x\right)$ exists at $x = a$. Then
${\lim}_{x \to a} f \left(x\right)$
$\quad \quad = {\lim}_{y \to g \left(a\right)} f \left({g}^{- 1} \left(y\right)\right)$

summary

Algebra of Limits

→  If sub-expressions are not evaluating to $0$ or $\infty$ then limit can be applied to sub-expressions.

→  If sub-expressions are evaluating to $0$ or $\infty$, then look for the forms of $\frac{0}{0}$.

Limit of Sum or Difference
»  Limit distributes over Addition and Subtraction
when value is not $\infty - \infty$
→  ${\lim}_{x \to a} \left[f \left(x\right) \pm g \left(x\right)\right]$$= {\lim}_{x \to a} f \left(x\right) \pm {\lim}_{x \to a} g \left(x\right)$

Limit of Product
»  Limit distributes over multiplication
when value is not $\infty \times 0$
→  ${\lim}_{x \to a} \left[f \left(x\right) \times g \left(x\right)\right]$$= {\lim}_{x \to a} f \left(x\right) \times {\lim}_{x \to a} g \left(x\right)$

Limit of Quotient
»  Limit distributes over division
when value is not $0 \div 0$ or $\infty \div \infty$
→  ${\lim}_{x \to a} \left[f \left(x\right) \div g \left(x\right)\right]$$= {\lim}_{x \to a} f \left(x\right) \div {\lim}_{x \to a} g \left(x\right)$

Limit of Exponent
»  Limit distributes over exponent
when value is not ${\infty}^{0}$ or ${0}^{0}$
→  ${\lim}_{x \to a} {\left[f \left(x\right)\right]}^{n}$$= {\left[{\lim}_{x \to a} f \left(x\right)\right]}^{n}$

Limit of Root
»  Limit distributes over root
when value is not ${\infty}^{0}$ or ${0}^{0}$
→  ${\lim}_{x \to a} {\left[f \left(x\right)\right]}^{\frac{1}{n}}$$= {\left[{\lim}_{x \to a} f \left(x\right)\right]}^{\frac{1}{n}}$

Change of Variable in a Limit
»  variable can be substituted
when value is not any of the forms of $\frac{0}{0}$
→  ${\lim}_{x \to a} f \left(x\right)$ $= {\lim}_{y \to g \left(a\right)} f \left({g}^{- 1} \left(y\right)\right)$

Outline