Why Your Limits Get Marked Wrong for Missing Justification

In high-school calculus and most first-year university calculus courses, you are expected to show not only what the answer is, but how you got it. For limits, this means:

• Writing the algebraic manipulation (factoring, cancelling, rationalizing, etc.) you used to simplify the expression.
• Stating the rule, theorem, or property that lets you make the step (for example, the limit laws, the Squeeze Theorem, or L’Hôpital’s Rule—always with their conditions met).
• Substituting values only when it is justified to do so (i.e., after simplifying the expression to remove an indeterminate form).

A common misconception is that if you can see the answer in your head or get it on a calculator, you can just write the answer. But in formal math courses, the grading is based on your reasoning process as much as your result. This is especially true on written assignments, WeBWorK with written parts, and all hand-marked exams.

Suppose you are asked to evaluate:

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$

If you simply write “4” as the answer, this will almost always lose marks for missing justification. Here is what a fully justified solution looks like:

Step 1: Factor the numerator.

$$\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2}$$

Step 2: State that for $x \neq 2$, you can cancel $x-2$:

$$= x + 2 \text{ for } x \neq 2$$

Step 3: Now, since the simplified expression $x+2$ is continuous at $x=2$, you can substitute:

$$\lim_{x \to 2} (x+2) = 2+2 = 4$$

Step 4: State the justification: “Since $x+2$ is continuous at $x=2$, the limit equals the value at 2.”

What counts as justification here?

• Factoring and algebra steps are shown.
• The cancellation is only done for $x \neq 2$.
• The property used (continuity) is stated.

If you skip straight to “4” or even just write “= 4” without any steps, you are not showing the reasoning, so you lose marks.

Some limits require a specific theorem to justify the answer. For example, if you use L’Hôpital’s Rule, you must:

• Check and state that the limit is in an indeterminate form (0/0 or ∞/∞).
• Write out the derivatives.
• State that you are applying L’Hôpital’s Rule.

Example:

$$\lim_{x \to 0} \frac{\sin x}{x}$$

If you just write “1”, you lose marks. Here’s what’s needed:

1. Check the form: $\sin 0 = 0, 0 = 0$ so this is $0/0$.

2. State the rule: “By L’Hôpital’s Rule (since the limit is 0/0),”

3. Differentiate numerator and denominator: $\lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1$

4. Conclusion: “Therefore, the limit is 1.”

If you use the Squeeze Theorem, you must write the inequalities you are squeezing between, show the limits of the bounding functions, and state that the Squeeze Theorem applies.

If a limit is so straightforward that it’s direct substitution (no indeterminate form, no algebra needed), then in some courses, you can just substitute:

For example:

$$\lim_{x \to 3} (2x+1) = 2(3)+1 = 7$$

Here, the function is continuous everywhere, and substitution is valid. However, in many university courses, you are still expected to write at least one line showing the substitution, not just “7”.

But: If the question explicitly says “state the answer only,” or if it is a multiple-choice question, then justification may not be needed. Always check the instructions.

A common trap is to write only the final algebraic step but not mention the rule. For example, writing:

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

without any explanation. In a high-school course, this might get partial credit if the teacher is lenient, but in first-year university calculus, it is usually not enough. You must say how you know it is 1 (e.g., “By the standard limit result” or “By L’Hôpital’s Rule”).

Another case: Using L’Hôpital’s Rule when the conditions do not apply. For example, trying to use it on a limit that is not 0/0 or ∞/∞. This will lose full marks, even if the answer is correct, because the justification is invalid.

Ask yourself:

1. Did I write every algebraic step, not just the answer?
2. Did I state the rule or property used (limit law, continuity, factoring, theorem)?
3. Did I check that the conditions for the rule are met (e.g. indeterminate form for L’Hôpital, correct inequalities for Squeeze Theorem)?
4. If I substituted a value, was it after resolving any indeterminate form?

If you can answer yes to all these, your justification is probably enough for a high-school or first-year calculus course.

• Check your course’s marking scheme or ask your instructor about expectations for written work.
• Look at the solutions provided by your textbook or instructor: how much detail do they show for limits?
• When in doubt, write more steps and name your rules. It is rare to lose marks for too much clear justification, but very common to lose them for too little.

If you want to build better habits for showing your reasoning, you can practice rewriting your solutions as if you’re explaining them to someone else. If you need extra help, Learn4Less can provide feedback, but you can make progress on your own by focusing on the structure above.

You are capable of giving full, accepted justifications—no special tricks are needed. The key is to make your reasoning visible, not just your answer.