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MathCalculus

Select Each Limit Law Used to Justify the Computation: lim (x³ + 3x² + 5)/(4 − 7x)

Selecting each limit law means naming the rule that licenses every step: the quotient law for the fraction, the sum and difference laws to split the top and bottom, the constant multiple and power laws for the terms, and the basic limits lim x = a and lim c = c. Applying them gives −3.

Question

In the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).

  1. lim⁡x→1x3+3x2+54−7x\displaystyle\lim_{x\to 1}\frac{x^{3}+3x^{2}+5}{4-7x}

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Evidence boundary

This page works the limit published as exercise 84 in the OpenStax Calculus Volume 1 §2.3 exercise set, whose instruction is to use the limit laws and to justify each step by naming the appropriate limit law(s). The laws listed follow that section: sum, difference, constant multiple, product, quotient, power, root, and the basic limit results lim x = a and lim c = c. The order in which the laws are applied here is one valid order, not the only one. The page does not claim the problem comes from any other worksheet, textbook or answer key, and it does not extend the computation to a general family of limits.