Evaluate the Limit

Problem

(lim_x→0)((5*x3+8*x2)/(3*x4−16*x2))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in 0/0

2. Factor out the greatest common factor, x2 from both the numerator and the denominator.

(x2*(5*x+8))/(x2*(3*x2−16))

3. Simplify the expression by canceling the common x2 terms, provided x≠0

(5*x+8)/(3*x2−16)

4. Evaluate the limit by substituting x=0 into the simplified expression.

(5*(0)+8)/(3*(0)2−16)

5. Calculate the final numerical value.

8/(−16)=−1/2

Final Answer

(lim_x→0)((5*x3+8*x2)/(3*x4−16*x2))=−1/2

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