How do you find the limit in the following question?

Question

#limxrarr∞#

#(x^3-2x^2)^(1/3) + (x^3-x^2)^(1/3) - (8x^3+4x^2)^(1/3)#

Ezekiel Alexander · Accepted Answer

#-4/3#

We use the binomial expansion for general index #n# :

#(1+z)^n = 1+nz+(n(n-1))/(2!)z^2+...#

for each of the three terms

#(x^3-2x^2)^(1/3) = x(1-2/x)^(1/3)# #qquad = x(1-1/3 times 2/x+O(1/x^2))# #qquad = x-2/3+O(1/x)#

#(x^3-x^2)^(1/3) = x(1-1/x)^(1/3) # #qquad = x(1-1/3 times 1/x+O(1/x^2))# #qquad = x-1/3+O(1/x)#

#(8x^3+4x^2)^(1/3) = 2x(1+1/(2x))^(1/3) # #qquad = 2x(1+1/3 times 1/(2x)+O(1/x^2))# #qquad = 2x+1/3+O(1/x)#

Combining these, we get

#(x^3-2x^2)^(1/3) + (x^3-x^2)^(1/3) - (8x^3+4x^2)^(1/3)# #qquad = [x-2/3+O(1/x)]+[x-1/3+O(1/x)]# #qquad -[2x+1/3+O(1/x)]# #qquad = -4/3+O(1/x)#

Thus, the limit of this expression as #x to oo# is #-4/3#

Luke Alvarez · Answer

undefined To find the limit in a given question, you need to evaluate the function as the input approaches a specific value. This can be done by direct substitution, factoring, rationalizing, or using algebraic manipulation. If direct substitution results in an indeterminate form (such as 0/0 or ∞/∞), you may need to apply limit theorems, such as L'Hôpital's rule or the squeeze theorem. Additionally, you can use properties of limits, such as sum, difference, product, and quotient rules, to simplify the expression before evaluating the limit.

How do you find the limit in the following question?

#limxrarr∞# #(x^3-2x^2)^(1/3) + (x^3-x^2)^(1/3) - (8x^3+4x^2)^(1/3)#

#limxrarr∞#

#(x^3-2x^2)^(1/3) + (x^3-x^2)^(1/3) - (8x^3+4x^2)^(1/3)#