Limit of a Function

Question 1

How to calculate a limit?

Answer

To calculate a limit, replace the variable with the value to which it tends to.

Example: Calculate the limit of $ f(x) = 2x $ when $ x $ tends to $ 1 $ written $ \lim_{x \to 1} f(x) $ is to calculate $ 2 \times 1 = 2 $ so $ \lim_{x \to 1} f(x) = 2 $.

In some cases, the result is undetermined (indeterminated limits, see below).

Question 2

How to calculate limits with 0 and $ \infty $ infinity?

Answer

Limit calculations usually use mathematical forms with values 0 or infinity (positive or negative), except indeterminated forms, calculations follow the rules:

$$ +\infty + \infty = +\infty $$	$$ -\infty - \infty = -\infty $$
$$ +\infty - \infty = ? $$	$$ -\infty + \infty = ? $$
$$ 0 + \infty = +\infty $$	$$ 0 - \infty = -\infty $$
$$ + \infty + 0 = +\infty $$	$$ - \infty + 0 = -\infty $$
$$ \pm k + \infty = +\infty $$	$$ \pm k - \infty = -\infty $$
$$ + \infty \pm k = +\infty $$	$$ - \infty \pm k = -\infty $$
$$ +\infty \times +\infty = +\infty $$	$$ +\infty \times -\infty = -\infty $$
$$ -\infty \times +\infty = -\infty $$	$$ -\infty \times -\infty = +\infty $$
$$ 0 \times +\infty = ? $$	$$ 0 \times -\infty = ? $$
$$ +\infty \times 0 = ? $$	$$ -\infty \times 0 = ? $$
$$ k \times +\infty = +\infty $$	$$ k \times -\infty = -\infty $$
$$ -k \times +\infty = -\infty $$	$$ -k \times -\infty = +\infty $$
$$ \frac{ +\infty }{ +\infty } = ? $$	$$ \frac{ +\infty }{ -\infty } = ? $$
$$ \frac{ -\infty }{ +\infty } = ? $$	$$ \frac{ -\infty }{ -\infty } = ? $$
$$ \frac{ 0 }{ +\infty } = 0 $$	$$ \frac{ 0 }{ -\infty } = 0 $$
$$ \frac{ +\infty }{ 0 } = +\infty $$	$$ \frac{ -\infty }{ 0 } = -\infty $$
$$ \frac{ +\infty }{ k } = +\infty $$	$$ \frac{ -\infty }{ k } = -\infty $$
$$ \frac{ +\infty }{ - k } = -\infty $$	$$ \frac{ -\infty }{ - k } = +\infty $$
$$ \frac{ k }{ +\infty } = 0^+ $$	$$ \frac{ k }{ -\infty } = 0^- $$
$$ \frac{ -k }{ +\infty } = 0^- $$	$$ \frac{ -k }{ -\infty } = 0^+ $$
$$ \frac{ 0 }{ 0 } = ? $$	$$ \frac{ k }{ k } = 1 $$
$$ \frac{ k }{ 0 } = + \infty $$	$$ \frac{ -k }{ 0 } = - \infty $$
$$ \frac{ 0 }{ k } = 0 $$	$$ \frac{ 0 }{ -k } = 0 $$
$$ (\pm k)^0 = 1 $$	$$ 0^{\pm k} = 0 $$
$$ 1^{\pm k} = 1 $$	$$ (\pm k)^1 = (\pm k) $$
$$ +\infty^0 = ? $$	$$ -\infty^0 = ? $$
$$ 0^{+\infty} = 0 $$	$$ 0^{-\infty} = 0 $$

With $ k> 0 $ a positive non-zero real constant.

The ? represent indeterminate forms.

Question 3

What are the indeterminate forms?

Answer

The indeterminate forms that appear when calculating limits are:

$$ \frac{0}{0} $$	0 divided by 0
$$ \frac{\pm\infty}{\pm\infty} $$	infinity divided by infinity
$$ 0 \times \pm\infty $$ or $$ \pm\infty \times 0 $$	0 multiplicated by infinity
$$ +\infty - \infty $$ or $$ -\infty + \infty $$	difference between infinity
$$ 0^0 $$	0 power 0
$$ \pm\infty^0 $$	infinity power 0
$$ 1^{\pm\infty}$$	1 power infinity

Question 4

How to calculate an indeterminate form?

Answer

Several methods related to limit calculations are possible.

1 - Factorize (using the dCode factorisation expression tools for example)

2 - Use the Hospital Rule (in cases of form $ 0/0 $ or $ \infty / \infty $: if $ f $ and $ g $ are 2 functions defined on the interval $ [a, b[ $ and differentiable in $ a $, and such that $ f(a) = g(a) = 0 $, then if $ g'(a) \ne 0 $: $$ \lim_{x \to a^+} \frac{f(x)}{g(x)} = \frac{f (a)}{g (a)} $$

3 - Use the dominant term rule (in the case of addition of polynomials and when the variable tends to infinity): the limit of a polynomial is the limit of its term of highest power.

4 - Calculate the asymptotes to deduce the limit values

5 - Transform the expression (using remarkable identities or extracting elements from the roots, etc.)

Limit of a Function

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Answers to Questions

How to calculate a limit?

How to calculate limits with 0 and \( \infty \) infinity?

What are the indeterminate forms?

How to calculate an indeterminate form?

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