I need to prove that the following limits exist but i have to use the definition of a limit.(the definition being that lim[(x,y)tends towards(a,b)]f(x)=L if for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |(x,y)-(a,b)| < δ, we have |ƒ(x) − L| < ε) I know how to show that a limit does not exist as well as how to find the limit by substitution but these have me stumped.The functions are:
1) lim [(x,y)tends to(0,1)]f(x) where f(x)= (x+y)/(y^2-x^2) By substitution this limit should equal 1 and I therefore need to show that there exists an e and d such that if sqrt(x^2+y^2-2y+1)<d then the absolute value of f(x)-1<e but I don't know how to derive this statement.Any ideas?
2) lim[(x,y)tends to(4,-2)]f(x) where f(x)=cuberoot(y^3+2x) by substitution this should be 0 but once again I don't know how to derive the statement that if sqrt[(x-4)^2+(y+2)^2]<d then the absolute value of f(x)<e
and finally
3) lim[(x,y)tends to (0,0)]f(x) where f(x)=x^4sin(x^2/y) for this one I know that the absolute value of sin(x^2/y)<=1 therefore absolute value of f(x)<=x^4 (for y is not 0) is this correct? then x^4< (sqrt(x^2+y^2))^4 so choosing d=e^4 we have the proof? I think I did something wrong here so if someone could please just check it for me?
Any help would be greatly appreciated. Thanks in advance.