To find the limit as x approaches positive infinity of the given expression, we can simplify the expression before taking the limit:
(√(1+x) - 3)/(1 + ∛x)
To simplify, we can first multiply the numerator and denominator by the conjugate of the numerator:
= (√(1+x) - 3)/(1 + ∛x) * (√(1+x) + 3)/(√(1+x) + 3)
Now, let's multiply out the numerator:
= (1+x - 9)/(1 + ∛x)
= x - 8 / x
Now we can simplify the expression using the limit properties:
lim x->+∞ (x - 8) / x
Since the degree of the numerator is the same as the degree of the denominator, we can divide the leading coefficients to find the limit:
= lim x->+∞ (x/x - 8/x)
= lim x->+∞ (1 - 8/x)
As x approaches positive infinity, the fraction 8/x goes to 0, so the limit simplifies to:
= 1 - 0
= 1
Therefore, the limit as x approaches positive infinity of (√(1+x) - 3)/(1 + ∛x) is equal to 1.
To find the limit as x approaches positive infinity of the given expression, we can simplify the expression before taking the limit:
(√(1+x) - 3)/(1 + ∛x)
To simplify, we can first multiply the numerator and denominator by the conjugate of the numerator:
= (√(1+x) - 3)/(1 + ∛x) * (√(1+x) + 3)/(√(1+x) + 3)
Now, let's multiply out the numerator:
= (1+x - 9)/(1 + ∛x)
= x - 8 / x
Now we can simplify the expression using the limit properties:
lim x->+∞ (x - 8) / x
Since the degree of the numerator is the same as the degree of the denominator, we can divide the leading coefficients to find the limit:
= lim x->+∞ (x/x - 8/x)
= lim x->+∞ (1 - 8/x)
As x approaches positive infinity, the fraction 8/x goes to 0, so the limit simplifies to:
= 1 - 0
= 1
Therefore, the limit as x approaches positive infinity of (√(1+x) - 3)/(1 + ∛x) is equal to 1.