2 questions!!!
1.
Limit X approaching A
(X^1/3-a^1/3)/ x-a
2.
LiMIT x approaching 0
(1/3+x – 1/3) /x
On the first, would it help to write the denominator (x-a) as the difference of two cubes ((x^1/3 cubed - a^1/3 cubed)
second. use LHopitals rule. Take the derivative of the numerator, derivative of the denominator.
what if u never learned the lhopitals rule
All you have to do is to take the derivative of both numerator and denominator. That's L'Hopital's rule
If you never learned L'Hopital's rule, you should derive it first. Suppose that (a) = 0
Lim x--> a f(x)/g(x) =
Lim h--->0 f(a+h)/g(a+h) =
Lim h--->0
(f(a+h)- (a+h) - g(a)) =
Lim h--->0
[(f(a+h)- f(a))/h] /(g(a+h) - g(a))/h] =
The limit of a quotient is the quotient of the limits provided both limits exists and the quotient of the limits also exists.
1. consider the question to look like this
lim (x^(1/3) - a^(1/3) ) / ( (x^1/3)^3 - (a^1/3)^3) as x ----> a
=lim (x^1/3 - a^1/3)/[ (x^1/3) - a^1/3)(x^(2/3) +x^1/3 a^1/3 + a^2/3) ] as x--->a
= lim 1/ (x^1/3) - a^1/3)(x^(2/3) +x^1/3 a^1/3 + a^2/3) as x --> a
= 1/(a^2/3 + a^1/3 a^1/3 + a^2/3)
= 1/(3 a^(2/3)
2. without L'Hopital's rule
lim (1/(3+x) - 1/3 )/x , as x-->0
= lim [(3 - 3 - x)/(3(3+x)) / x
= lim [ -x/(3(3+x))/x
= lim -1/(3(3+x)) as x-->0
= -1/9
1. To find the limit as x approaches a of (x^(1/3) - a^(1/3))/(x-a), you can rewrite the denominator as the difference of cubes. The difference of cubes formula states that a^3 - b^3 = (a-b)(a^2 + ab + b^2).
So, in this case, you can rewrite (x-a) as (x^(1/3))^3 - (a^(1/3))^3, which becomes (x^(1/3) - a^(1/3))(x^(2/3) + (x^(1/3))(a^(1/3)) + a^(2/3)).
Now, you can cancel out the common factor of (x^(1/3) - a^(1/3)) from the numerator and the denominator, leaving you with (x^(2/3) + (x^(1/3))(a^(1/3)) + a^(2/3)).
To find the limit, you can now directly substitute a into the expression, which gives you a^(2/3) + (a^(1/3))(a^(1/3)) + a^(2/3) = 2(a^(2/3)). So, the limit as x approaches a of (x^(1/3) - a^(1/3))/(x-a) is 2(a^(2/3)).
2. To find the limit as x approaches 0 of ((1/3) + x - (1/3))/x, you can simplify the expression first. Combine the (1/3) terms, which gives you ((1/3) - (1/3))/x = 0/x = 0.
So, the limit as x approaches 0 of ((1/3) + x - (1/3))/x is 0.
If you have not learned L'Hopital's rule, you can still find the limit by using basic algebraic simplifications. L'Hopital's rule is a shortcut method that uses derivatives to evaluate limits, but it is not the only way to find limits. You can apply basic algebraic manipulations to simplify the expression and then evaluate the limit using direct substitution or other methods.