Using Limit Rules

Suppose li...

Question

Using Limit Rules

Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)

(We assume the denominator is nonzero.)

(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)

= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)

= ((2)(1) − (−5)) / (1 + 7)² = 7/64

Accepted Answer

Welcome back, everyone. Identify the limit law applied in the equality below. Limit as X approaches -2 of 5PX minus 3QX divided by 2 PFX plus 6 raises the power of 4 equals limit as x approaches -2 of 5% of X minus 3 QX divide. By limit as x approaches -2 of 2% of X plus 6 raises the power of 4. We are given 4 answer choices. A says some law for limits, different law for limits, C product law for limits and the quotient law for limits. So let's analyze each option. First of all, we have a which says. Some law for limits and we have to recall that it simply states that if we have limit as X approaches some point A of a sum between F of X and G of X, we can simply distribute the limit and we can write this as a sum of two separate limits. Limit SX approaches A of F of X plus limit. As x approaches a of G of X, and now if we carefully analyze our limit well, we did not apply this rule, specifically, we don't have any sum to consider other than the one in the denominator but we have. The 4th power outside, right? So it's definitely not the some law for limits. Now, let's consider option B, the first law, basically it is the same property as the Some low for limits, but now we are replacing the addition with subtraction. We can say that if we have a difference between two functions, we can simply distribute the limit and get two limits that are being subtracted, right? In this case, we have limit of P X minus VQ X in the numerator, but we did not distribute the limit for each function, right? So it is not. Different law for limits. Now C says product law for limits, and now the product law says that we take limit as X approaches some point A of a product F of X multiplied by G of X, and once again we can distribute the limit. As as X approaches A of F of X multiplied by limit as X approaches A of G of X. In our function, we do not have any products, so essentially we can exclude option C, it is not correct, right? We did not apply the product law. This means that the correct answer should be quotient law for limits, and now let's understand that quotient law for limits allows us to take limit as x approaches some point A of the ratio between F of X and G of X. If we have a ratio L divided by M, where L and M are our limits such that M is not equal to 0, right, because division by 0 is undefined, then this property allows us to distribute the limit as follows. Limit as x approaches A of F of X divided by. Limit as x approaches a of G of X, right, as long as m is not equal to 0, so we can see that. This is the quotient law for limits, and this is exactly what we have in our expression. We have distributed the limit for the numerator and the denominator separately without affecting anything else, which means that the correct answer to this problem corresponds to the answer choice the quotient flow for limits. Thank you for watching.

Key Concepts

Limit of a Function

Limit Laws

Continuity and Non-zero Denominator

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