Tool for calculating limits of mathematical functions. A limit is defined by the value of a function when its variable approaches a given value.

Limit of a Function - dCode

Tag(s) : Functions

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To calculate a limit, replace the variable with the value to which it tends/approaches to (close neighborhood).

__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 (indeterminate limits, see below) and can suggest the existence of an asymptote.

Limit calculations usually use mathematical forms with values 0 or infinity (positive or negative), except indeterminate 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.

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 multiplied by infinity |

$$ +\infty - \infty $$ or $$ -\infty + \infty $$ | difference between infinities |

$$ 0^0 $$ | 0 power 0 |

$$ \pm\infty^0 $$ | infinity power 0 |

$$ 1^{\pm\infty} $$ | 1 power infinity |

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.)

The sine and cosine functions, tending to $ \pm \infty $, do not admit a limit because they are periodic (reproducing an infinite pattern) and therefore do not tend towards a finite value, nor towards an infinity. Their limit is indefinite, but sometimes noted $ \pm 1 $ (not recommended).

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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