Evaluate the Limit limit as x approaches 0 of 1/(x^2(x+7))

Problem

(lim_x→0)(1/(x2*(x+7)))

Solution

1. Analyze the behavior of the numerator and denominator as x approaches 0

2. Evaluate the limit of the numerator, which is a constant 1

3. Evaluate the limit of the denominator by substituting x=0 into the expression x2*(x+7)

4. Observe that as x→0 x2 is always positive and approaches 0 while (x+7) approaches 7

5. Determine that the denominator x2*(x+7) approaches 0 from the positive side (0.

6. Conclude that since the numerator is a positive constant and the denominator is a positive value approaching 0 the fraction grows without bound.

Final Answer

(lim_x→0)(1/(x2*(x+7)))=∞

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