You want to find lim_{x→2} (x² + 3x)/(x − 2). You apply the quotient law and get (4 + 6)/(2 − 2) = 10/0. What should you conclude?
AThe limit is infinity, since 10/0 = ∞
BThe limit does not exist, since the quotient law produces an undefined result
CThe quotient law cannot be applied here because the denominator's limit is 0; further analysis is required
DYou should apply L'Hôpital's rule directly to get lim = 10/1 = 10
The quotient law states lim(f/g) = L/M only when M ≠ 0. When M = 0, the quotient law simply does not apply — you cannot draw any conclusion from it. The result 10/0 is not '∞'; it signals that this case requires different analysis. Here lim(x²+3x) = 10 ≠ 0 and lim(x−2) = 0, which indicates vertical asymptote behavior — but you must determine this through further work, not by mechanically writing '10/0 = ∞.'
Question 2 Multiple Choice
A student evaluates lim_{x→3} (x²−9)/(x−3) by substituting x = 3, gets 0/0, and concludes the limit doesn't exist. A second student factors to get lim_{x→3}(x+3) = 6. Who is correct?
AThe first student; substituting x = 3 gives 0/0 which means the limit is undefined
BNeither; a 0/0 form always means further limit laws must be applied iteratively
CThe second student; 0/0 is an indeterminate form signaling the quotient law fails, not that the limit doesn't exist
DBoth are correct; 0/0 and 6 are equivalent in limit notation
0/0 is an indeterminate form — it means the quotient law fails and further algebraic work is needed, not that the limit is undefined or doesn't exist. The second student correctly factors the numerator as (x−3)(x+3), cancels (x−3) (valid since x ≠ 3 in a limit), and applies direct substitution to (x+3), getting 6. The indeterminate form is a signal to look for hidden algebraic structure, not a dead end.
Question 3 True / False
For any polynomial p(x), the limit as x approaches any real number a can be found by direct substitution: lim_{x→a} p(x) = p(a).
TTrue
FFalse
Answer: True
This follows directly from the limit laws. A polynomial is built from sums and products of constants and powers of x. The limit laws say limits distribute over sums and products. The foundational limits lim_{x→a} c = c and lim_{x→a} x = a hold by definition. Applying the limit laws repeatedly through the polynomial's structure reduces everything to p(a). Direct substitution for polynomials is not magic — it is the limit laws working automatically.
Question 4 True / False
Limit laws are the definitions of what limits mean — they establish how limits of sums, products, and quotients are computed.
TTrue
FFalse
Answer: False
Limit laws are theorems, not definitions. The limit is defined independently (via epsilon-delta or as the value a function approaches). The limit laws are then proved from that definition: if you can get f close to L and g close to M, you can prove rigorously that f+g gets close to L+M. The laws are derived results that make computation convenient. This distinction matters because the laws have conditions (like M ≠ 0 for the quotient law) that make no sense if they were definitions.
Question 5 Short Answer
Explain why the quotient law fails when the denominator's limit is zero, and what this failure tells you about how to proceed.
Think about your answer, then reveal below.
Model answer: The quotient law requires M ≠ 0 because division by zero is undefined, and when M = 0, the ratio f/g can behave in any number of ways as x approaches a: it might approach a finite value (0/0 indeterminate form, requiring algebraic simplification), blow up to ±∞ (vertical asymptote), or fail to exist. The failure of the quotient law is a diagnostic signal — not a final answer — that tells you which case you're in and that a different technique is needed: factoring and canceling, conjugate multiplication, L'Hôpital's rule, or the squeeze theorem.
Recognizing when limit laws fail is as important as knowing what they say. 'The law doesn't apply' is informative: it tells you that you're in a case requiring special treatment. Students who mechanically write '0/0' as an answer have misunderstood the quotient law; students who recognize it as an indeterminate form know they need a different strategy.