lim as x->infinity
[(x^3+x^2)^1/3] - [(x^3-x^2)^1/3]
To find the limit as x approaches infinity of the given expression, [(x^3+x^2)^(1/3)] - [(x^3-x^2)^(1/3)], you can use the property of limits. Specifically, when dealing with fractions, we can take the limit of the numerator and denominator separately.
Let's begin by simplifying the expression.
The first term, (x^3+x^2)^(1/3), can be rewritten as [(x^3(1+x)^2)^(1/3)]. Using the exponent properties, we can split the product inside the parentheses: (x^3)^(1/3) * (1+x)^(2/3). The cube root of x^3 simplifies to x, so the first term becomes x * (1+x)^(2/3).
Similarly, the second term, (x^3-x^2)^(1/3), can be rewritten as [(x^2(x-1))^(1/3)]. Using the same exponent properties, we can split the product: (x^2)^(1/3) * (x-1)^(1/3). The cube root of x^2 simplifies to x^(2/3), so the second term becomes x^(2/3) * (x-1)^(1/3).
Now we have the expression in a simplified form: x * (1+x)^(2/3) - x^(2/3) * (x-1)^(1/3).
To find the limit as x approaches infinity, we can consider the highest power of x in the expression. In this case, it is x^(2/3) in the second term.
Since x^(2/3) approaches infinity as x increases, the second term dominates the first term in the expression. Therefore, we can ignore the first term and focus on finding the limit of just the second term.
The limit of x^(2/3) * (x-1)^(1/3) as x approaches infinity can be determined by considering the dominant term, which is x^(2/3). We can rewrite the expression as x^(2/3) * [ (x-1)^(1/3) / x^(2/3) ]. As x approaches infinity, both x and (x-1) also approach infinity.
Now, let's focus on the ratio inside the square brackets, (x-1)^(1/3) / x^(2/3). We can rewrite this expression as ( (x-1) / x )^(1/3) * ( 1 / x )^(2/3).
When we take the limit, both factors inside the brackets approach 1/1, resulting in 1^(1/3) * 1^(2/3) = 1 * 1 = 1.
Therefore, the limit as x approaches infinity of the given expression is 1.
In summary:
lim as x->infinity [(x^3+x^2)^(1/3)] - [(x^3-x^2)^(1/3)] = 1