How to compute the limit of a function (in Python, using SymPy)

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Task

In mathematics, we write

to refer to the value that $f$ approaches as $x$ gets close to $a$, called “the limit of $f(x)$ as $x$ approaches $a$.”

How can we use software to compute such limits?

Solution

This answer assumes you have imported SymPy as follows.

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from sympy import *                   # load all math functions
init_printing( use_latex='mathjax' )  # use pretty math output

Here we define a simple mathematical formula, $\frac{x^2-x-2}{x-2}$, and compute the limit as $x$ approaches 2. We use SymPy’s built-in limit function, which takes the formula $f(x)$, the variable $x$, and the value $a$.

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var( 'x' )
formula = ( x**2 - x - 2 ) / ( x - 2 )
limit( formula, x, 2 )

$\displaystyle 3$

You can also compute one-sided limits. For instance, the limit of $\frac{\vert x\vert}{x}$ is $1$ as $x$ approaches 0 from the right, but it is $-1$ as $x$ approaches 0 from the left.

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limit( abs(x)/x, x, 0, "-" )  # "-" means from the left

$\displaystyle -1$

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limit( abs(x)/x, x, 0, "+" )  # "+" means from the right

$\displaystyle 1$

Content last modified on 24 July 2023.

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Contributed by Nathan Carter (ncarter@bentley.edu)