CRL 2.1 Limits MrG 2011.0928

3706 days ago by calcpage123

plot(2*x+1,xmin=-1,xmax=1,aspect_ratio=1) 
       
plot(x^3) 
       
f(x)=(2*x^2-1)/(3*x^2+1);f(x) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, x^{2} - 1}{3 \, x^{2} + 1}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, x^{2} - 1}{3 \, x^{2} + 1}
plot(f(x),xmax=30) 
       
for x in range(100,1000,100): print(f(x).n()) 
       
0.666611112962901
0.666652777893518
0.666660493850023
0.666663194451678
0.666664444447407
0.666665123458219
0.666665532880590
0.666665798611563
0.666665980795893
0.666611112962901
0.666652777893518
0.666660493850023
0.666663194451678
0.666664444447407
0.666665123458219
0.666665532880590
0.666665798611563
0.666665980795893
a=plot(1/x,xmin=0.1,xmax=10) b=plot(1/x,xmin=-10,xmax=-0.1) show(a+b,aspect_ratio=1) 
       
limit(1/x,x=Infinity) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0
limit(1/x,x=-Infinity) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0
limit(1/x,x=0,dir='right') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}+\infty
\newcommand{\Bold}[1]{\mathbf{#1}}+\infty
limit(1/x,x=0,dir='left') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\infty
\newcommand{\Bold}[1]{\mathbf{#1}}-\infty
limit(f(x),x=infinity) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2}{3}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2}{3}
f(x)=(x^2-4)/(x-2);f(x) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{2} - 4}{x - 2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{x^{2} - 4}{x - 2}
plot(f(x),xmin=1.999,xmax=2.001) 
       
limit(f(x),x=2) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}4
\newcommand{\Bold}[1]{\mathbf{#1}}4
f(2) 
       
Traceback (click to the left of this block for traceback)
...
RuntimeError: power::eval(): division by zero
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_12.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZigyKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmphY3Km2/___code___.py", line 3, in <module>
    exec compile(u'f(_sage_const_2 )
  File "", line 1, in <module>
    
  File "expression.pyx", line 3652, in sage.symbolic.expression.Expression.__call__ (sage/symbolic/expression.cpp:16183)
  File "/home/medlock/sage-4.7.1/local/lib/python2.6/site-packages/sage/symbolic/callable.py", line 451, in _call_element_
    return SR(_the_element.substitute(**d))
  File "expression.pyx", line 3503, in sage.symbolic.expression.Expression.substitute (sage/symbolic/expression.cpp:15547)
RuntimeError: power::eval(): division by zero
limit(f(x),x=2,dir='left') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}4
\newcommand{\Bold}[1]{\mathbf{#1}}4
limit(f(x),x=2,dir='right') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}4
\newcommand{\Bold}[1]{\mathbf{#1}}4
g(x)=e^x;g(x) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}e^{x}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{x}
plot(g(x),xmin=-10,xmax=10) 
       
limit(g(x),x=-infinity) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}0
\newcommand{\Bold}[1]{\mathbf{#1}}0
limit(g(x),x=infinity) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}+\infty
\newcommand{\Bold}[1]{\mathbf{#1}}+\infty
g(5) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
limit(g(x),x=5,dir='left') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
limit(g(x),x=5,dir='right') 
       
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
limit(g(x),x=5) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{5}
bool(g(5)==limit(g(x),x=5)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
\newcommand{\Bold}[1]{\mathbf{#1}}{\rm True}
plot([e^x,ln(x),x],xmin=0,xmax=2,aspect_ratio=1) 
       
a=plot(e^x,xmin=-3,xmax=1) b=plot(x,xmin=-3,xmax=2) c=plot(ln(x),xmin=0,xmax=e) show(a+b+c,aspect_ratio=1)