Tool for calculating limits of mathematical functions. Quickly and accurately calculate the value of a function when its variable approaches a given value.
Limit of a Function - dCode
Tag(s) : Functions
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Limit of a Function
Limits Calculator
Answers to Questions (FAQ)
What is the limit of a function? (Definition)
In mathematics, the limit of a function is a value that the function approaches as the variable approaches a particular value.
Why use function limits?
Function limits are essential in the analysis of functions, to understand their behavior at particular points, such as the extreme values (minimum, maximum) of their domain of definition or towards their points of discontinuity.
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.)
What is the difference between a left limit and a right limit?
A left bound (limit by the left) refers to the value that the function approaches as the variable approaches the target value from lower values.
A right bound (limit by the right) refers to the value that the function approaches as the variable approaches the target value from higher values.
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.
Source code
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- Limits Calculator
- What is the limit of a function? (Definition)
- Why use function limits?
- 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?
- What is the difference between a left limit and a right limit?
- How to calculate limits of trigonometric functions like sine and cosine?
- How to show step by step calculations?
