/set approximation exact
/set fr 2
limit(x^2-4,2)
0
limit((x^3-4x)/(2x^2+3x),0)
-4/3
limit(x^3/(x+1)^2,-1)
-infinity
limit((x+1)^2(x-1)/(x^3+1),-1)
0
limit((x^3-2x^2+x)/(2x^3+x^2-2x),0)
-1/2
limit((x^2+2x+3)/(x-1)^2,1)
+infinity
limit((x^4-4x^3+x^2)/(x^3+x^2+x),0)
0
limit((x^3+x^2+x+1)/(x^4+x^2-2),-1)
-1/3
#limit((x+1)^2/(2-x),2)
# undefined
limit((x-2)/(x^2-3x+2),2)
1
limit((3x+2x^-1)/(x+4x^-1),0)
1/2
limit((x^2-3x+2)/(x^2-2x),2)
1/2
limit(1/(1-x)-3/(1-x^3),1)
-1
limit((x^2-x-2)/(x^2-2x),2)
3/2
limit((x^2+5)/(x^2-3),2)
9
limit((3x^4-4x^3+1)/(x-1)^2,1)
6
limit((3x+6)/(x^3+8),-2)
1/4
limit((x+1)/(x-1),2)
3
limit((x^3+3x^2+2x)/(x^2-x-6),-2)
-2/5
limit((x^2-2x+1)/(x^3-x),1)
0
limit((x^2+7x-44)/(x^2-6x+8),4)
15/2
#limit((x^2-4)/(x^2-3x+2),1)
# undefined
limit((x^3-5x+4)/(x^3-1),1)
-2/3
limit((x^2-4)/(x-2),2)
4
limit((x^2-4)/(x^2-3x+2),2)
4
limit(1/(x^2-1)-2/(x^4-1),1)
1/2
limit((x-3)/(x^2-5x+6),3)
1
limit((x^2-1)/(2x^2+1),infinity)
1/2
limit((x^3+x^2-4)/(2x^3+x+11),-infinity)
1/2
limit((3x^2+2x-1)/(x^3-x+2),infinity)
0
limit((x^3)/(x^2+2)-x,infinity)
0
limit((x^2+3x-4)/(3x^2-2x+5),infinity)
1/3
limit((x(x-1)(x-2))/(x^2+6x-9),infinity)
+infinity
limit((sqrt(x^2+9))/(x+3),infinity)
1
limit(((x^2+x-1)/(2x^2-x+1))^3,infinity)
1/8
limit((x^2+2x+1)/(5x),infinity)
+infinity
limit((x^3+x^4-1)/(2x^5+x-x^2),-infinity)
0
limit((sqrt(x^2+1)+x)^2/(cbrt(x^6+1)),infinity)
4
limit((x^6+7x^4-40)/(1-x-5x^7),-infinity)
0
limit(((x+1)(x-2))/(3x^2+6x-5),infinity)
1/3
limit(sqrt(x^2+1)/(x),infinity)
1
limit(((3x^2+2x+1)/(x^2-3x+2))^4,infinity)
81
limit((5x^3-x^2+x)/(1-x-3x^2),-infinity)
+infinity
limit((1+x-3x^3)/(1+x^2+3x^3),infinity)
-1
limit(((x^3-8)/(x^4+16))^10,-infinity)
0
limit(((x+3)(x+4)(x+5))/(x^4+x-11),infinity)
0
limit((8x-2x^5+x^6)/(11x+5x^3+3x^5),-infinity)
-infinity
limit((x^3)/(2x^2-1)-x^2/(2x+1),infinity)
1/4
limit(x^2-(x^4-1)/(x^2-2),infinity)
-2
limit(((x-1)^100*(6x+1)^200)/(3x+5)^300,infinity)
3117982410208
limit((root(x^5,4)+root(x^3,5)+root(x^8,6))/(cbrt(x^4+2)),infinity)
1
limit((x^2(2x+1)(3x-2))/(2x^2(5x-8)(x+6)),-infinity)
3/5
limit(((2x^8+8x^6+6x^4)/(4x^8-x^6+12x^4))^5,infinity)
1/32
limit(((2x-3)^20*(3x+2)^30)/(2x+1)^50,-infinity)
191751.0592
limit((sqrt(1+2x)-1)/(3x),0)
1/3
limit((sqrt(x+1)-sqrt(1-x))/(x),0)
1
limit((x-sqrt(x))/(sqrt(x)),0)
-1
limit((sqrt(x-1)-2)/(x^2-25),5)
1/40
#limit((sqrt(x+6)-3)/(x^3-5x^2+3x+9),3)
# undefined
limit((3-sqrt(x))/(27-sqrt(x^3)),9)
1/27
limit((cbrt(1+x)-cbrt(1-x))/(x),0)
2/3
limit((x^(2/3)-1)/(x^(3/5)-1),1)
10/9
/assume positive
limit((1 − (x^(1 / z))) / (1 − (x^(1 / y))), 1)
y / z
/assume unknown
limit(sqrt(x-2)-sqrt(x),infinity)
0
limit(sqrt(x^2+x)-x,infinity)
1/2
limit(sqrt(x^2+x)-1,-infinity)
+infinity
limit(sqrt(x-3)-sqrt(x),infinity)
0
limit(sqrt(x)*(sqrt(x-3)-sqrt(x)),infinity)
-3/2
limit(x(sqrt(x^2+1)-x),infinity)
1/2
limit(x(sqrt(x^2+1)-x),-infinity)
-infinity
limit(sqrt(x^2+1)-x,infinity)
0
limit(sqrt(x^2+1)-x,-infinity)
+infinity
limit((sqrt(x+2)-sqrt(2))/(x),infinity)
0
limit((sqrt(x+5)-sqrt(5))/(sqrt(x)-5),infinity)
1
limit((sqrt(x^2+9)-sqrt(x^2-9))/(6x),infinity)
0
limit((sqrt(x-1)-2x)/(x-7),infinity)
-2
limit((sqrt(x)-6x)/(3x+1),infinity)
-2
limit((sqrt(x^2+1)+sqrt(x))/(root(x^3+x,4)-x),infinity)
-1
limit((sqrt(x^2+1)+sqrt(x))/(root(x^2+1,4)-x),infinity)
-1
limit((cbrt(x)-2*sqrt(x^3))/(root(x^5,4)+x*sqrt(x)),infinity)
-2
#limit((sqrt(x^2+1)-cbrt(x^2+1))/(2*root(x^4+1,4)-root(x^4+1,5)),-infinity)
# 1/2
limit((1+1/x)^(3x),infinity)
e^3
limit((1+1/x^2)^(3x),infinity)
1
limit((1+1/x^2)^(3x-4),infinity)
1
limit((1+1/(5x))^(2x),infinity)
e^(2/5)
limit((1+1/(5x))^(2x+6),infinity)
e^(2/5)
limit((1+7/(3x))^(x),infinity)
e^2 * cbrt(e)
limit((1+7/(3x))^(x-1),infinity)
e^2 * cbrt(e)
limit((1-1/3x)^x,infinity)
1 / cbrt(e)
limit((1-5/x)^x,infinity)
1 / e^5
limit(((x-1)/(x+1))^x,infinity)
1 / e^2
limit(((x+6)/(x+5))^x,infinity)
e
limit(((x-3)/(x))^(x/2),infinity)
1 / (e * sqrt(e))
limit(((7x+10)/(1+7x))^(x/3),infinity)
e^(3/7)
limit(((2x-1)/(2x+1))^(x),infinity)
1 / e
limit(((6+4x)/(2+4x))^(-2x),infinity)
1 / e^2
#limit(((6+4x)/(2+4x))^(3-2x),infinity)
# 1 / e^2
limit(((x+5)/(x+4))^(2x),infinity)
e^2
#limit(((x+5)/(x+4))^(2x-1),infinity)
# e^2
limit(((2x+5)/(2x))^(3x),infinity)
e^7 * sqrt(e)
#limit(((2x+5)/(2x))^(3x-7),infinity)
# e^7 * sqrt(e)
limit(((2x+1)/(2x-3))^(3x),infinity)
e^6
limit(((x-1)/(x+3))^(-4x),infinity)
e^16
#limit(((x-1)/(x+3))^(5-4x),infinity)
# e^16
limit(((x)/(x+1))^(x),infinity)
1 / e
limit(((3x-2)/(3x+1))^(2x),infinity)
1 / e^2
limit(((2x-5)/(2x-2))^(4x^2),infinity)
0
limit(((3x+6)/(3x-1))^(x^2),infinity)
+infinity
limit(((x^2+2x+2)/(x^2+3))^(x),infinity)
e^2
limit((1+2x)^(1/x),0)
e^2
limit((1+5x)^(1/8x),0)
e^(5/8)
limit(ln(((3x+1)/(3x-5))^(-x)),infinity)
-2
limit(ln(((3x+1)/(3x-5))^(2-x)),infinity)
-2
limit(x*(ln(x+3)-ln(x)),infinity)
3
limit(x*(ln(x)-ln(x+2)),infinity)
-2
limit((x+1)*(ln(x+1)-ln(x)),infinity)
1
limit(sin(10x)/(10x),0)
1
limit(sin(3x)/(2x),0)
3/2
limit(tan(8x)/(x),0)
8
limit(sin(3x)/sin(5x),0)
3/5
limit(tan(5x)/sin(4x),0)
5/4
limit(tan(5x)/tan(6x),0)
5/6
limit(tan(x)/(3x),0)
1/3
limit((1-cos(x))/(x),0)
0
#limit((x)/(sqrt(1-cos(x))),0)
# undefined
limit(sin(x)/(x^3),0)
+infinity
limit((1-cos(2x))/(x*sin(x)),0)
2
limit((sin(x/2)^3)/(x^3),0)
1/8
limit((sin(4x)+sin(7x))/(sin(3x)),0)
11/3
#limit((sqrt(1-cos(x^2)))/(1-cos(x)),0)
# sqrt(2)
limit((tan(x)-sin(x))/(x^3),0)
1/2
limit(x*sin(pi/x),infinity)
pi
limit((x^3+1)/sin(x+1),-1)
3
#limit(abs(sin(x))/(x),0)
# undefined
limit((tan(x-1))/(sqrt(x)-1),1)
2
limit((sqrt(cos(x))-1)/(sin(x)^2),0)
-1/4
limit((sin(3x)+sin(5x))/sin(2x),0)
4
limit((cos(x)-cos(x)^3)/(x^2),0)
1
limit(x*cot(2x),0)
1/2
#limit(sqrt((1-cos(x))/(x^2)),0)
# sqrt(2)/2
limit((sqrt(1-tan(x))-sqrt(1+tan(x)))/sin(2x),pi)
-1/2
limit(ln(x)/ln(sin(x)),0)
1
limit(ln(sin(2x))/ln(sin(x)),0)
1
limit((3^x-1)/(6^x-1),0)
ln(3) / ln(6)
limit(x(2^(1/x)-1),infinity)
ln(2)
limit((e^x+x)^(1/x),0)
e^2
limit((e^((1+x)/(1-x)))/2,infinity)
1 / (2e)
limit((e^x-e^(-x))/(2),infinity)
+infinity
limit((e^x+e^(-x))/(e^x-e^(-x)),-infinity)
-1
limit(ln(1+x)/(x),0)
1
limit(ln(1+3x)/(x),0)
3
limit((ln(x)-1)/(x-e),e)
1 / e
limit(ln((x-1)/(x+1)),-infinity)
0
limit(1/x*ln(sqrt((1+x)/(1-x))),0)
1
limit((ln(x^x)-x)/(2-ln(x^2)),e)
-e / 2
limit((1-3^x)/(sin(3x)),0)
-ln(3) / 3
limit(2^x*sin(2pi*x),-infinity)
0
limit((sin(2x))/(sqrt(x+3)-sqrt(3)),0)
4 * sqrt(3)
#limit(x^2*(1-cos(1/x)),infinity)
# 1/2
limit(cos(x)^(1/x),0)
1
limit((1-sin(x))/(pi-2x),pi/2)
0
limit((sin(x)-cos(x))/cos(2x),pi/4)
-1 / sqrt(2)
#limit(sin(sqrt(x+1))-sin(sqrt(x)),infinity)
# undefined
#limit(tan(x)^tan(2x),pi/4)
# 1 / e
limit(asin(x)/(x),0)
1
limit(asin(5x)/(3x),0)
5/3
#limit(e^(-x)/asin(1/x),-infinity)
# -infinity
limit(acos(sqrt(x^2+x)-x),infinity)
pi / 3
limit(acot(x)/(x^2-x),infinity)
0
limit(sqrt(pi/2-atan(1/(x-1)^2)),1)
0
limit((x^3-2x^2+x)/(x^2-1),1)
0
limit((x^3-9x)/(x^4-3x^3-x+3),3)
9/13
limit((cbrt(8x)-2x)/(root(x,4)-x),1)
16/9
limit((sqrt(x+1)-sqrt(2))/(x^2-1),1)
1 / (sqrt(2) * 4)
limit((5^x-1)/(x),0)
ln(5)
limit((3x^3-3)/(3^x-3),1)
3 / ln(3)
limit((e^x-e^(-x))/(x),0)
2
#limit((e^x-1)/(x^2),0)
# undefined
limit((1-2*cos(x))/(pi-3x),pi/3)
-1 / sqrt(3)
limit((sin(x))/(e^x-1),0)
1
limit(atan(3x)/asin(2x),0)
3/2
limit((e^x-e^(-x))/(2x-sin(x)),0)
2
limit((x^3+pi*x)/sin(3x),0)
pi / 3
limit(ln(1+4x)/(3^x-1),0)
4 / ln(3)
limit(sin(4x)/ln(1+sin(x)),0)
4
limit((3*ln(1-2x))/(2*atan(3x)),0)
-1
limit((atan(x)+x^2)/(2^(3x)-3^(2x)),0)
1 / (3 * ln(2) - 2 * ln(3))
limit(atan(x-pi/2)/(pi-2x),pi/2)
-1/2
limit((e^(3x)-e^(-2x))/(2*asin(x)-sin(x)),0)
5
limit(ln(cos(3x))/atan(4x),0)
0
limit(((1+x)^2-(1+2x))/(x^2+4x^3),0)
1
limit((x^2+3x+2)^2/(x^3-3x-2),-1)
-1/3
limit((1-cos(x)^4)/(4x^2),0)
1/2
limit((1-cos(x))/(2x*sin(x)),0)
1/4
limit((x-sin(x))/(e^x-e^(-x)-2x),0)
1/2
limit((e^(3x)-3x-1)/(sin(x)^2),0)
9/2
limit((x^3)/(x-atan(x)),0)
3
limit((1-sin(x)/x)^(1/ln(x)),0)
e^2