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Limit of a Function

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

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Limit of a Function -

Tag(s) : Functions

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# Limit of a Function

## Limits Calculator

 Direction Automatic Right limit (from larger values, tends to X+) Left limit (from smaller values, tends to X-)

### How to calculate a limit?

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.

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

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.

### 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 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

### 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 calculate limits of trigonometric functions like sine and cosine?

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).

### 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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