To evaluate the limit as x approaches negative infinity of ((1+x)^5)/∛(-x^15), we first simplify the expression.
As x approaches negative infinity, the terms (1+x)^5 and -x^15 both tend towards infinity. However, the denominator ∛(-x^15) becomes -x^5 because the cube root function preserves the sign.
Therefore, the expression ((1+x)^5)/∛(-x^15) simplifies to ((1+x)^5)/-x^5.
Now, expand (1+x)^5 using the binomial theorem:
(1+x)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5
Therefore, ((1+x)^5)/-x^5 becomes:
(1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5)/- x^5
Now, divide each term by -x^5:
(1/x^5 + 5/x^4 + 10/x^3 + 10/x^2 + 5/x + 1)
As x approaches negative infinity, all terms except the 1/x^5 term approach 0. Therefore, the limit of the expression ((1+x)^5)/∛(-x^15) as x approaches negative infinity is 1.
To evaluate the limit as x approaches negative infinity of ((1+x)^5)/∛(-x^15), we first simplify the expression.
As x approaches negative infinity, the terms (1+x)^5 and -x^15 both tend towards infinity. However, the denominator ∛(-x^15) becomes -x^5 because the cube root function preserves the sign.
Therefore, the expression ((1+x)^5)/∛(-x^15) simplifies to ((1+x)^5)/-x^5.
Now, expand (1+x)^5 using the binomial theorem:
(1+x)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5
Therefore, ((1+x)^5)/-x^5 becomes:
(1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5)/- x^5
Now, divide each term by -x^5:
(1/x^5 + 5/x^4 + 10/x^3 + 10/x^2 + 5/x + 1)
As x approaches negative infinity, all terms except the 1/x^5 term approach 0. Therefore, the limit of the expression ((1+x)^5)/∛(-x^15) as x approaches negative infinity is 1.