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.)
How to show step by step calculations?
The dCode limit calculator does not apply school methods but bit-by-bit calculation, so the calculation steps are very different and are not displayed.
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