Evaluate Using L'Hospital's Rule limit as x approaches 8 of x^3e^(-x^2)

Problem

(lim_x→8)(x3)*e(−x2)

Solution

1. Identify the type of limit by substituting the value x=8 into the expression.

2. Evaluate the expression directly since it is a continuous function at x=8

3. Substitute x=8 into the term x3

8=512

4. Substitute x=8 into the term e(−x2)

e(−(8))=e(−64)

5. Combine the results to find the limit. Note that L'Hospital's Rule is not applicable here because the limit does not result in an indeterminate form like 0/0 or ∞/∞

(lim_x→8)(x3)*e(−x2)=512*e(−64)

Final Answer

(lim_x→8)(x3)*e(−x2)=512*e(−64)

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