# How to calculate a limit of a function in python

To find the limit of a mathematical function in python, we can use limit() function of the sympy library.

**limit(y,x,x0,s)**

- The first argument (y) is the
**functionf (x)**whose limit is to be calculated - The second parameter (x) is the reference
**variable**( argument ) - The third parameter (x0) is the
**accumulation point**

**oo**= + infinity**-oo**= - infinity**0**= zero**n**= number

- The fourth parameter (s) is used to calculate the
**lateral limits**of a point**'+'**=right limit**'-'**= left limit

The function **limit()** calculates the limit of the function f (x) as x approaches x_{0}.

$$ \lim_{x \rightarrow x_0 } f(x) = l $$

**What's a limit of a function? **The limit of a function f(x) at X_{0} point in its domain, if it exists, is the value that the function f(x) approaches as its argument approaches X_{0}. The notation of a limit is as follows: $$ \lim_{x \rightarrow x_0 } f(x) = l $$ We can read "*the limit of f(x) as x approaches x _{0} is L*".

The limit() function must be imported from the sympy library using the command from sympy import limit.

## Examples

__Example 1__

This script calculates the limit of the function 1/x as x approaches zero.

from sympy import limit, Symbol

x = Symbol('x')

y=1/x

limit(y, x, 0)

The function returns to output

oo

The limit of the function 1 / x as x approaches zero is more infinite (oo).

$$ \lim_{x \rightarrow 0 } \frac{1}{x} = \infty $$

__Example 2__

This script calculates the limit of the 1/x function as x approaches + infinite.

from sympy import limit, oo, Symbol

x = Symbol('x')

y=1/x

limit(y, x, oo)

The oo symbol (+ infinity) must be imported from sympy.

The output of the function is

0

The limit of the function 1 / x for x tending to +∞ is zero.

$$ \lim_{x \rightarrow \infty } \frac{1}{x} = 0 $$

__Example 3
__

This script calculates the limit of the function x^{2} as x approaches - infinite.

from sympy import limit, oo, Symbol

x = Symbol('x')

y=x**2

limit(y, x, -oo)

The output of the function is

oo

The limit of the function is + infinite.

$$ \lim_{x \rightarrow - \infty } x^2 = \infty $$

__Example 4__

This script calculates the limit of the x^{2} as x approaches 4.

from sympy import limit, oo, Symbol

x = Symbol('x')

y=x**2

limit(y, x, 4)

The output of the function is

16

The limit of the function for x tending to 4 is 16.

$$ \lim_{x \rightarrow 4 } x^2 = 16 $$

__Example 5 (lateral limit)
__

This script calculates the lateral boundary of the 1 / x function with x tending towards zero from the left.

from sympy import limit, Symbol

x = Symbol('x')

y=1/x

limit(y, x, 0, '-')

The function returns in output

-oo

The limit of the function 1 / x as x tending to zero from the left is minus infinite (-oo).

$$ \lim_{x \rightarrow 0^- } \frac{1}{x} = - \infty $$

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