Math 115 (Summer 2025): Written HW 1

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On this page:

• Using square brackets and $\pm \infty$ in intervals
• Writing the endpoints of intervals in the wrong order
• Forgetting parentheses
• Being lazy and not writing $\displaystyle\lim_{x\to a}$ when you should
• Writing $\displaystyle\lim_{x\to a}$ when you shouldn’t be

On this page, I give you some common notation mistakes that calculus students make on Written HW 1 that WILL be penalized.

Using square brackets and $\pm \infty$ in intervals

[example] Description: When you use interval notation, NEVER put a square bracket next to $\pm \infty$.

Why? Putting a square bracket next to $\pm \infty$ means you are treating $\pm \infty$ AS A NUMBER but $\pm \infty$ is really NOTATION to mean “we extend infinitely to the left/right”.

Examples:

1. Write the inequality $x \le 3$ in interval notation.
2. Write the inequality $x > -1$ in interval notation.
3. Write $x \neq 5$ in interval notation. [/example]

[redbox] [redtitle]WRONG[/redtitle]

1. $[-\infty, 3]$.
2. $[2, \infty]$.
3. $[-\infty, 5) \cup (5, \infty]$ [/redbox]

[definition] [deftitle]CORRECT[/deftitle]

1. $(-\infty, 3]$.
2. $[2, \infty)$.
3. $(-\infty, 5) \cup (5, \infty)$ [/definition]

Writing the endpoints of intervals in the wrong order

[example] Description: When you use interval notation, you ALWAYS write the “smaller” endpoint on the left and the larger one on the “right” (the quotation marks are because of $\pm \infty$ as endpoints). [/example]

[redbox] [redtitle]WRONG[/redtitle]

1. $(2, -5]$
2. $(1, -\infty)$.
3. $(\infty, 0)$ [/redbox]

[definition] [deftitle]CORRECT[/deftitle]

1. $[-5, 2)$
2. $(-\infty, 1)$.
3. $(0, \infty)$ [/definition]

Forgetting parentheses

[example] Description: ALWAYS double-check to make sure you aren’t missing parentheses.

Why? Order of operations and this pretty much speaks for itself.

Example: Compute the difference quotient for

[/example]

[redbox] [redtitle]WRONG[/redtitle]

[/redbox]

[definition] [deftitle]CORRECT[/deftitle]

[/definition]

Being lazy and not writing $\displaystyle\lim_{x\to a}$ when you should

[example] Description: Being too lazy to write $\lim_{x\to a}$ because “the grader should understand what I mean.”

Why? Because doing so leads to nonsensical mathematical statements like $x + 5 = 3 + 5$ as in below. Also, if there are lots of variables, it may not be obvious which variable you are taking the limit in.

Example:

1. Compute $\displaystyle\lim_{x\to 3} (x + 5)$.
2. Compute $\displaystyle\lim_{h\to 0} (2xh + x^2h)$. [/example]

[redbox] [redtitle]WRONG[/redtitle]

1. $x + 5 = 3 + 5 = 8$.
   • The whole point of $x$ being a variable is to NOT assume its a specific number. So $x + 5 \neq 3 + 5$.
2. $2xh + x^2h = 0 + 0 = 0$.
   • In addition to the above, it is not obvious if we are letting $x \to 0$ or $h \to 0$. [/redbox]

[definition] [deftitle]CORRECT[/deftitle]

1. $\displaystyle\lim_{x\to 3} (x + 5) = 3 + 5 = 8$
2. $\displaystyle\lim_{h\to 0} (2xh + x^2h) = 2x\cdot 0 + x^2\cdot 0 = 0$. [/definition]

Writing $\displaystyle\lim_{x\to a}$ when you shouldn’t be

[example] Description: After computing a limit, sometimes people will write $\lim_{x\to a}$ as part of their final answer.

Why? This communicates the idea that you want to take the limit again… not what you meant to do.

Example: Compute

[/example]

[redbox] [redtitle]WRONG[/redtitle]

Since

our final answer must be

(This is wrong because this is saying “take the limit of $8$ as $x \to 3$ when what was really meant was “the limit of $x + 5$ as $x \to 3$ is $8$). [/redbox]

[definition] [deftitle]CORRECT[/deftitle]

Since

our final answer must be

[/definition]