CEGEP Calculus I — Indeterminate Forms and L'Hospital's Rule

All 4 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Recognizing an Indeterminate Form

   For each limit below, carry out direct substitution and write down the form it produces. State whether that form is indeterminate. If it is not indeterminate, give the value of the limit (or write “does not exist”, with a reason).

   1. ${\displaystyle {lim}_{x\to 1}\frac{x^3-1}{x^2-1}}$
   2. ${\displaystyle {lim}_{x\to 2^{-}}\frac{x+4}{x-2}}$
   3. ${\displaystyle {lim}_{x\to \infty }\frac{3x^2-x}{5x^2+7}}$
   4. ${\displaystyle {lim}_{x\to 0}\frac{\cos x}{x^2}}$
   5. ${\displaystyle {lim}_{x\to \infty }\frac{\ln x}{e^x}}$
   6. ${\displaystyle {lim}_{x\to 0}\frac{x^2}{\sin x+3}}$
2. Q2L'Hospital's Rule for Quotient Forms

   Evaluate each limit with L'Hospital's Rule. State the form you get from substitution before each application of the rule, so that it is clear the rule is allowed each time.

   1. ${\displaystyle {lim}_{x\to 0}\frac{e^{3x}-3x-1}{x^2}}$
   2. ${\displaystyle {lim}_{x\to \infty }\frac{\ln (1+5x)}{\ln (1+2x)}}$
   3. ${\displaystyle {lim}_{x\to \pi /2}\frac{\cos x}{x-\frac{\pi }{2}}}$
3. Q3Choosing Between Algebra and L'Hospital's Rule

   For each limit below, decide whether it is settled more efficiently by algebraic manipulation, by L'Hospital's Rule, or equally well by either. Justify the choice in one sentence — saying what the rejected method would cost you — and then evaluate the limit by the method you chose.

   1. ${\displaystyle {lim}_{x\to 4}\frac{\sqrt{x}-2}{x-4}}$
   2. ${\displaystyle {lim}_{x\to \infty }\frac{\sqrt{9x^2+4x}}{2x+7}}$
   3. ${\displaystyle {lim}_{x\to 0}\frac{e^{2x}-1}{\sin 3x}}$
4. Q4Synthesis — drawing on several topics in this set

   Let $$g(x)=\frac{\sqrt{x+9}-3}{x^2+2x}.$$

   1. State the domain of $g$, then evaluate ${\displaystyle {lim}_{x\to 0}g(x)}$ twice — once algebraically and once with L'Hospital's Rule — and confirm the two agree.
   2. Determine ${\displaystyle {lim}_{x\to -2}g(x)}$. Explain first why L'Hospital's Rule must not be used here, and say what wrong answer it would have produced.
   3. Determine ${\displaystyle {lim}_{x\to \infty }g(x)}$, naming the form and the method you chose.

The 3 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

An indeterminate form is a report, not an answer

Q1 gives you six limits and asks for one thing first: substitute, and write down the form that comes out. That instruction is the whole topic in miniature. Substitution either produces a number, in which case you are finished, or it produces a symbol — 0/0, ∞/∞, something over 0 — and the symbol tells you what kind of work is still owed. It never tells you the answer.

Calling a form indeterminate is a precise claim: expressions of that shape can approach any real number, or ±∞, or nothing at all, so the form on its own rules out no outcome. That is why 0/0 does not "equal 1" and ∞/∞ does not "equal 1". Two things are racing to zero, or two things are running off to infinity, and which one wins is a question about the functions, not about the symbol.

What this course does with the rule, and what Calculus II does

Worth knowing before you revise, because it is the difference between studying the right thing and studying twice as much as you need to.

The split is not arbitrary tidiness. Competency 0M02 names the quotient forms; the rewrite forms sit in 0M03, alongside the improper integrals that use the same limit machinery. So a limit of the shape 1^∞ turning up in your Calculus I revision is a sign you have wandered into next term's material, not a sign you have missed something.

Stating the form before every single application

Q2 asks you to evaluate three limits with the rule and to state the form you get from substitution before each application, so that it is clear the rule is allowed each time. Read that as a specification of what a complete answer looks like, rather than as a request for extra writing.

The rule is a conditional statement, with a hypothesis on one side of it and a conclusion on the other, and the hypothesis is about the quotient at that stage. One of Q2's limits does not resolve on a single application: you differentiate, look at the new quotient, and find yourself facing an indeterminate form again. Applying the rule a second time is entirely legitimate there — but it is legitimate because you checked the new form, not because the rule was allowed a moment ago. A second application on an unchecked quotient is the commonest way a correct-looking solution turns out to be wrong.

Q2's three limits are chosen to make the checking habit worth having: one where the ∞/∞ form is built from two logarithms rather than two polynomials, one that needs the rule more than once, and one where the quotient is exactly a derivative read off its own definition — so the value can be confirmed against something you already know, which is the best kind of check there is.

The decision the devis actually assesses

Q3 is the sheet that earns this topic its place. For each limit you decide whether it is settled more efficiently by algebra, by L'Hospital's Rule, or equally well by either, justify the choice in one sentence saying what the rejected method would have cost you, and only then evaluate. The performance criterion in the ministerial devis is application pertinente — a pertinent application of the rule, not merely a correct one. Getting the right number by the laborious route is a partial answer to the question being asked.

Notice the shape of Q3's instruction — say what the rejected method would cost you. A choice defended with "this is faster" is not yet a justification. Defended with "the rule is legal here but returns the same ∞/∞ form, whereas dividing by the dominant power finishes in one line", it is. And a limit that really is settled equally well either way is a legitimate answer to give, provided you say so and show that the two routes agree.

One function, three points, three different situations

Q4 closes the set by putting everything on a single function, g(x) = (√(x + 9) − 3)/(x² + 2x), and asking about it in three places. Start where the question starts, with the domain: the root needs a non-negative radicand and the denominator factors into two zeros, so the domain arrives as a union of intervals, and the excluded points are exactly the ones parts (a) and (b) then ask about. Domain first is not a formality here — it is what tells you those two excluded points are not the same kind of thing.

Part (a) asks for one limit twice, algebraically and with the rule, and to confirm that the two agree. The algebraic route needs the conjugate, since a difference of a root and a constant has no factor to pull out; the rule route needs the derivative of a square root. Two independent methods landing on the same value is the cheapest verification in calculus, and it is worth doing deliberately while you are still learning which method to reach for.

Part (c) sends the same function to infinity and asks you to name the form and the method you chose. It is the ∞/∞ form again, and that wording — the method you chose — is a reminder that by this point choosing is the assessed skill. Compare how fast the numerator grows against the denominator, and let the comparison pick the route before you differentiate anything.

Getting the most out of it

Write the form on its own line, every time

Before any manipulation, and before every re-application of the rule, write down what substitution gives. It costs one line, it is what the questions ask for, and it is the only thing standing between you and applying the rule to a quotient that was never indeterminate. Once writing the form is a habit, the misapplications stop happening on their own.

Decide before you differentiate

Look at the expression for a moment first: is there a common factor, a conjugate, a dominant power? Algebra usually wins on quotients built from polynomials and roots; the rule usually wins where exponentials, logarithms or trigonometric functions are mixed in. Ten seconds of deciding saves two applications of the rule and half a page of derivatives.

Check the value against something else

A second route, a numerical value at an input close to the target, or a growth-rate comparison — any of them will catch a slip. Where a limit can be done both ways, doing it both ways is not wasted effort while you are still building the judgement about which way to pick.

Common questions