from sympy import limit, Symbol, oo, sqrt, Rational, log, exp, cos, sin, tan, \
    pi, asin, together

"""
(*) in problem number means that the number is relative to the book "Anti-demidovich,
problemas resueltos, Ed. URSS"

"""

x = Symbol("x", real=True, unbounded=True)

def test_leadterm():
    assert (3+2*x**(log(3)/log(2)-1)).leadterm(x)==(3,0)

def sqrt3(x):
    return x**Rational(1,3)

def sqrt4(x):
    return x**Rational(1,4)

def test_Limits_simple_0():
    assert limit((2**(x+1)+3**(x+1))/(2**x+3**x),x,oo)==3  #175

def test_Limits_simple_1():
    assert limit((x+1)*(x+2)*(x+3)/x**3,x, oo)==1  #172
    assert limit(sqrt(x+1)-sqrt(x),x,oo)==0  #179
    assert limit((2*x-3)*(3*x+5)*(4*x-6)/(3*x**3+x-1),x,oo)==8  #Primjer 1
    assert limit(x/sqrt3(x**3+10),x,oo)==1  #Primjer 2
    assert limit((x+1)**2/(x**2+1),x,oo)==1  #181

def test_Limits_simple_2():
    assert limit(1000*x/(x**2-1),x,oo)==0  #182
    assert limit((x**2-5*x+1)/(3*x+7),x,oo)==oo  #183
    assert limit((2*x**2-x+3)/(x**3-8*x+5),x,oo)==0  #184
    assert limit((2*x**2-3*x-4)/sqrt(x**4+1),x,oo)==2  #186
    assert limit((2*x+3)/(x+sqrt3(x)),x,oo)==2  #187
    assert limit(x**2/(10+x*sqrt(x)),x,oo)==oo  #188
    assert limit(sqrt3(x**2+1)/(x+1),x,oo)==0  #189
    assert limit(sqrt(x)/sqrt(x+sqrt(x+sqrt(x))),x,oo)==1  #190

def test_Limits_simple_3a():
    a = Symbol('a', real=True)
    #issue 414
    assert together(limit((x**2-(a+1)*x+a)/(x**3-a**3),x,a)) == \
            (a-1)/(3*a**2)  #196

def test_Limits_simple_3b():
    h = Symbol("h")
    assert limit(((x+h)**3-x**3)/h,h,0)==3*x**2  #197
    assert limit((1/(1-x)-3/(1-x**3)),x,1)==-1  #198
    assert limit((sqrt(1+x)-1)/(sqrt3(1+x)-1),x,0)==Rational(3)/2  #Primer 4
    assert limit((sqrt(x)-1)/(x-1),x,1)==Rational(1)/2  #199
    assert limit((sqrt(x)-8)/(sqrt3(x)-4),x,64)==3  #200
    assert limit((sqrt3(x)-1)/(sqrt4(x)-1),x,1)==Rational(4)/3  #201
    assert limit((sqrt3(x**2)-2*sqrt3(x)+1)/(x-1)**2,x,1)==Rational(1)/9  #202

def test_Limits_simple_4a():
    a = Symbol('a', real=True)
    assert limit((sqrt(x)-sqrt(a))/(x-a),x,a)==1/(2*sqrt(a))  #Primer 5
    assert limit((sqrt(x)-1)/(sqrt3(x)-1),x,1)==Rational(3)/2  #205
    assert limit((sqrt(1+x)-sqrt(1-x))/x,x,0)==1  #207
    assert limit(sqrt(x**2-5*x+6)-x,x,oo)==-Rational(5)/2  #213

def test_limits_simple_4aa():
    assert limit(x*(sqrt(x**2+1)-x),x,oo)==Rational(1)/2  #214

def test_Limits_simple_4b():
    #issue 412
    assert limit(x-sqrt3(x**3-1),x,oo)==0  #215

def test_Limits_simple_4c():
    assert limit(log(1+exp(x))/x,x,-oo)==0  #267a
    assert limit(log(1+exp(x))/x,x,oo)==1  #267b

def test_bounded():
    assert limit(sin(x)/x, x, oo) == 0 #216b
    assert limit(x*sin(1/x), x, 0) == 0 #227a

def test_f1a():
    h = Symbol("h")
    #issue 409:
    assert limit((sin(2*x)/x)**(1+x),x,0) == 2 #Primer 7

def test_f1a2():
    #issue 410:
    assert limit(((x-1)/(x+1))**x,x,oo) == exp(-2) #Primer 9

def test_f1b():
    m = Symbol("m")
    n = Symbol("n")
    h = Symbol("h")
    a = Symbol("a")
    assert limit(sin(x)/x,x,2) == sin(2)/2 #216a
    assert limit(sin(3*x)/x,x,0) == 3 #217
    assert limit(sin(5*x)/sin(2*x),x,0) == Rational(5)/2 #218
    assert limit(sin(pi*x)/sin(3*pi*x),x,0) == Rational(1)/3 #219
    assert limit(x*sin(pi/x),x,oo) == pi #220
    assert limit((1-cos(x))/x**2,x,0) == Rational(1,2) #221
    assert limit(x*sin(1/x),x,oo) == 1 #227b
    assert limit((cos(m*x)-cos(n*x))/x**2,x,0) == ((n**2-m**2)/2) #232
    assert limit((tan(x)-sin(x))/x**3,x,0) == Rational(1,2) #233
    assert limit((x-sin(2*x))/(x+sin(3*x)),x,0) == -Rational(1,4) #237
    assert limit((1-sqrt(cos(x)))/x**2,x,0) == Rational(1,4) #239
    assert limit((sqrt(1+sin(x))-sqrt(1-sin(x)))/x,x,0) == 1 #240

    assert limit((1+h/x)**x,x,oo) == exp(h) #Primer 9
    assert limit((sin(x)-sin(a))/(x-a),x,a) == cos(a) #222, *176
    assert limit((cos(x)-cos(a))/(x-a),x,a) == -sin(a) #223
    assert limit((sin(x+h)-sin(x))/h,h,0) == cos(x) #225

def test_f2a():
    assert limit(((x+1)/(2*x+1))**(x**2),x,oo) == 0 #Primer 8

def test_f2():
    assert limit((sqrt(cos(x))-sqrt3(cos(x)))/(sin(x)**2),x,0) == -Rational(1,12) #*184

def test_f3():
    a = Symbol('a', real=True)
    #issue 405
    assert limit(asin(a*x)/x, x, 0) == a