Evaluate the Limit

Problem

(lim_x→∞)((x2*(1−4*x3))/(x4*(2*x+1)))

Solution

1. Distribute the terms in the numerator and the denominator to simplify the expression.

(x2−4*x5)/(2*x5+x4)

2. Identify the highest power of x in the expression, which is x5

3. Divide every term in the numerator and the denominator by x5

((x2)/(x5)−(4*x5)/(x5))/((2*x5)/(x5)+(x4)/(x5))

4. Simplify the resulting fractions.

(1/(x3)−4)/(2+1/x)

5. Apply the limit as x→∞ noting that any term with x in the denominator approaches 0

(0−4)/(2+0)

6. Calculate the final value.

−2

Final Answer

(lim_x→∞)((x2*(1−4*x3))/(x4*(2*x+1)))=−2

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