Limit of a Function

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 indeterminated forms, calculations follow the rules:

With $ k > 0 $ a positive non-zero real constant.

The ? represent indeterminate forms.

The indeterminate forms that appear when calculating limits are:

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.