Tool for calculating limits of mathematical functions. A limit is defined by the value of a function when its varaible approaches a given value.
Limit of a Function - dCode
Tag(s) : Mathematics
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Limit of a Function
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Tool for calculating limits of mathematical functions. A limit is defined by the value of a function when its varaible approaches a given value.
Answers to Questions
How to calculate a limit?
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).
How to calculate limits with 0 and \( \infty \) infinity?
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.
What are the indeterminate forms?
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 |
How to calculate an indeterminate form?
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.)
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