Tasks:
The Limit Laws introduced in Section 5.3 (Theorem 5.5) summarize some of the operations we can do to simplify limit calculations.
Example 5.6 has a very simple example in which the limit of a rational function is evaluated -- you won't have to break every limit down like this. In fact Theorem 5.7 generalizes this for any rational function (and any point
𝑐
in its domain).
This discussion is also contained in this video:

Question:

Which of the following statements is not a Limit Law?

(Assume all limits exist.)

Question

Tasks:
The Limit Laws introduced in Section 5.3 (Theorem 5.5) summarize some of the operations we can do to simplify limit calculations. 
Example 5.6 has a very simple example in which the limit of a rational function is evaluated -- you won't have to break every limit down like this. In fact Theorem 5.7 generalizes this for any rational function (and any point 
𝑐
 in its domain).
This discussion is also contained in this video:

Question:

Which of the following statements is not a Limit Law?

(Assume all limits exist.)

BlackTom AI · Accepted Answer

All options concern limit laws, and we will evaluate each one carefully before deciding which is not a standard limit law.
Option 1: lim_{x→c} [f(x) + g(x)] = lim_{x→c} f(x) + lim_{x→c} g(x). This is a classic limit law: the limit of a sum equals the sum of the limits (assuming the limits exist, as stated). It is a valid and widely used property.
Option 2: lim_{x→c} [α f(x)] = α · lim_{x→c} f(x). This expresses linearity of limits with respect to constant multiples. Since α is a constant, pulling it out is legitimate, giving the limit of a scalar multiple as the scalar multiple of the limit. This is another standard limit law.
Option 3: lim_{x→c} [f(x) · g(x)] = [lim_{x→c} f(x)] · [lim_{x→c} g(x)]. This is the product law for limits: the limit of a product equals the product of the limits, provided each limit exists. It is a valid, commonly used rule in limit calculations.
Option 4: lim_{x→a} f(x) + lim_{x→b} f(x) = lim_{x→(a+b)} f(x). Here we are combining two separate limits at potentially different points a and b and equating them to the limit at a single, different point (a+b). This is not a standard limit law; limits at distinct points do not in general relate in this way, and the expression on the right does not correspond to a universal limit rule. The equality can easily fail depending on f and the points a, b. Therefore, this option is not a valid limit law.
In summary, Options 1, 2, and 3 align with established limit laws, while Option 4 contradicts the usual behavior of limits and is not a valid law.

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