MATH02, Limits and series
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Limits
Limit of a function
import sympy
#symypy.init_printing()
from sympy import symbols, limit, sin
x = symbols('x')
limit(sin(x) / x, x, 0)
OUTPUT : \(1\)
Derivatives using limit
import sympy
#symypy.init_printing()
from sympy import symbols, Function, limit, cos
f = Function('f')
x, h = symbols("x, h")
diff_limit = (f(x + h) - f(x))/h
limit(diff_limit.subs(f, cos), h, 0)
OUTPUT : \(- \sin{\left (x \right )}\)
Asymptotic behavior
import sympy
#symypy.init_printing()
from sympy import symbols, Function, limit, oo
f = Function('f')
x = symbols("x")
expr = (x**2 - 3*x) / (2*x - 2)
p = limit(expr/x, x, oo)
q = limit(expr - p*x, x, oo)
p, q
OUTPUT : \(\displaystyle \left( \frac{1}{2}, \ -1\right)\)
Series for an unspecified function
Maclaurin series
import sympy
#symypy.init_printing()
from sympy import symbols, Function, series
x = symbols("x")
f = Function("f")(x)
series(f, x, n=3)
OUTPUT : \(\displaystyle f{\left(0 \right)} + x \left. \frac{d}{d x} f{\left(x \right)} \right|_{\substack{ x=0 }} + \frac{x^{2} \left. \frac{d^{2}}{d x^{2}} f{\left(x \right)} \right|_{\substack{ x=0 }}}{2} + O\left(x^{3}\right)\)
Taylor series
import sympy
#symypy.init_printing()
from sympy import symbols, Function, series
x, x0 = symbols("x, {x_0}")
f = Function("f")(x)
f.series(x, x0, n=2)
OUTPUT : \(\displaystyle f{\left({x_0} \right)} + \left(x - {x_0}\right) \left. \frac{d}{d \xi_{1}} f{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}={x_0} }} + O\left(\left(x - {x_0}\right)^{2}; x\rightarrow {x_0}\right)\)
Approximation on taylor series
import sympy
#symypy.init_printing()
from sympy import symbols, Function, series
x, x0 = symbols("x, {x_0}")
f = Function("f")(x)
f.series(x, x0, n=2).removeO()
OUTPUT : \(f{\left({x_0} \right)} + \displaystyle \left(x - {x_0}\right) \left. \frac{d}{d \xi_{1}} f{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}={x_0} }}\)
Series for an specified function
Univariate
import sympy
#symypy.init_printing()
from sympy import symbols, cos, series
x = symbols("x")
cos(x).series(n=10)
OUTPUT : \(\displaystyle 1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} - \frac{x^{6}}{720} + \frac{x^{8}}{40320} + O\left(x^{10}\right)\)
Multivariate
import sympy
#symypy.init_printing()
from sympy import symbols, cos, sin, series
x, y = symbols("x, y")
expr = cos(x) / (1 + sin(x * y))
X = expr.series(x, n=2)
Y = expr.series(y, n=2)
X, Y
OUTPUT : \(\displaystyle \left( 1 - x y + O\left(x^{2}\right), \ \cos{\left(x \right)} - x y \cos{\left(x \right)} + O\left(y^{2}\right)\right)\)
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