Determine $\lim _{x \rightarrow-3} \frac{2 x^3-18 x}{x+3}$.
a) -36
b) 6
c) -6
d) 36
Answer (2)
∙ Recognize that direct substitution results in an indeterminate form 0 0 .
∙ Factor the numerator to obtain 2 x ( x − 3 ) ( x + 3 ) .
∙ Simplify the expression by canceling the ( x + 3 ) term, resulting in 2 x ( x − 3 ) .
∙ Evaluate the limit by substituting x = − 3 into the simplified expression: 36 .
Explanation
Problem Analysis We are asked to find the limit of the function x + 3 2 x 3 − 18 x as x approaches − 3 .
Indeterminate Form First, notice that if we directly substitute x = − 3 into the expression, we get − 3 + 3 2 ( − 3 ) 3 − 18 ( − 3 ) = 0 2 ( − 27 ) + 54 = 0 − 54 + 54 = 0 0 , which is an indeterminate form. This means we need to simplify the expression before we can evaluate the limit.
Factoring the Numerator To simplify, we can factor the numerator: 2 x 3 − 18 x = 2 x ( x 2 − 9 ) . Further, we can factor the quadratic term as a difference of squares: x 2 − 9 = ( x − 3 ) ( x + 3 ) . So, the numerator becomes 2 x ( x − 3 ) ( x + 3 ) .
Simplifying the Expression Now we can rewrite the original expression as: x + 3 2 x 3 − 18 x = x + 3 2 x ( x − 3 ) ( x + 3 ) As long as x = − 3 , we can cancel the ( x + 3 ) terms: x + 3 2 x ( x − 3 ) ( x + 3 ) = 2 x ( x − 3 )
Evaluating the Limit Now we can evaluate the limit by substituting x = − 3 into the simplified expression: x → − 3 lim 2 x ( x − 3 ) = 2 ( − 3 ) ( − 3 − 3 ) = 2 ( − 3 ) ( − 6 ) = 36
Final Answer Therefore, the limit of the function as x approaches − 3 is 36.
Examples
In physics, when analyzing the motion of an object, you might encounter a situation where the position function involves a rational expression that becomes indeterminate at a certain time. Evaluating the limit, as we did here, helps determine the object's velocity or acceleration at that specific instant, providing insights into its behavior even when direct substitution fails.
The limit of the function as x approaches -3 is 36, derived by factoring and simplifying the expression. This necessary simplification was prompted by the indeterminate form obtained with direct substitution. Therefore, the correct answer is option d) 36.
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