# Limits

Mathematics I · Differential Calculus · https://tryals.app/learn/mathematics-i/limits

## The Core Idea of Limits

The **limit** is the foundational concept of calculus. It describes the value a function **approaches** as its input nears a point, ignoring what happens **at** that point.

$$\lim_{x \to a} f(x) = L$$

As $x$ gets arbitrarily close to $a$ (without equalling it), $f(x)$ nears $L$. This examines the **neighborhood** of $a$, ignoring $a$ itself.

**One-sided limits** approach from one direction: $x \to a^-$ (left) and $x \to a^+$ (right). The two-sided limit **exists only when both one-sided limits exist and agree**.

## Indeterminate Forms

Limits shine on **indeterminate forms** like $\frac{0}{0}$. Consider $\lim_{x \to 2}\frac{x^2 - 4}{x - 2}$. Direct substitution yields $\frac{0}{0}$. Factoring and cancelling leaves $x + 2$, giving a limit of $4$. The function is undefined at $x = 2$, yet the limit exists.

| Pitfall | Reality |
| --- | --- |
| $\lim_{x \to a} f(x) = f(a)$ | Limits ignore $a$, can exist when $f(a)$ is undefined. |
| $\frac{0}{0}$ means no limit | It means algebra is needed; the limit often exists. |

Limits obey clean **algebra of limits** rules for sums, products, and quotients.

## Practice questions

5 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. If $\lim_{x \to a} f(x) = L$ then $f(a) = L$.

**Answer:** False

**Why:** False. The limit describes behaviour *approaching* $a$ and deliberately ignores the value there, $f(a)$ may differ from $L$, or not exist at all. The two coinciding is precisely the definition of continuity at $a$, which is why continuity has to be stated as a separate condition rather than following from the limit.

Page: https://tryals.app/practice/mathematics-i/limits/if-lim-x-a-f-x-l-then-f-a-l

### 2. Order the steps of evaluating the limit of $(x^{2} - 4)/(x - 2)$ as x approaches 2.

**Answer:**

1. Substitute x = 2 and obtain the indeterminate form 0/0
2. Factor the numerator as (x - 2)(x + 2)
3. Cancel the common factor x - 2, which is legitimate because x is never equal to 2
4. Substitute x = 2 into the simplified expression x + 2
5. Report the limit as 4

**Why:** Substitution is always the first move, and the 0/0 outcome is what licenses the algebra. The justification in step 3 matters: cancelling x - 2 changes the function AT x = 2, but the limit never looks there, so the two functions have the same limit.

Page: https://tryals.app/practice/mathematics-i/limits/order-the-steps-of-evaluating-the-limit-of-x-4-x-2-as-x

### 3. The sequence 0.9, 0.99, 0.999, … has limit 1. Does any term equal 1, and does that matter?

A. No term equals 1 and it does not matter, since 0.999… is the limit
B. The question is undecidable without specifying how many nines
C. The final term equals 1, which is why the limit is 1
D. No term equals 1, so 0.999… is slightly less than 1

**Answer:** A. No term equals 1 and it does not matter, since 0.999… is the limit

**Why:** The notation names the limit, not a term, and no number lies strictly between 0.999… and 1, which settles it. The persistent objection assumes the symbol means 'a very long term', and the whole content of a limit is that it means the value being approached.

Page: https://tryals.app/practice/mathematics-i/limits/the-sequence-0-9-0-99-0-999-has-limit-1-does-any-term-equal-1

### 4. The form $0/0$ does not mean a limit fails to exist. What does it actually indicate?

A. That the limit is indeterminate, so more work is needed
B. That the limit is zero, since the numerator vanishes
C. That the limit is one, since the two parts are equal
D. That the expression cannot be simplified any further

**Answer:** A. That the limit is indeterminate, so more work is needed

**Why:** Indeterminate means undecided, not undefined. Every derivative is a $0/0$ limit, so treating the form as failure would abolish differential calculus.

Page: https://tryals.app/practice/mathematics-i/limits/the-form-0-0-does-not-mean-a-limit-fails-to-exist-what-does-it

### 5. Sort each limit by its value.

**Answer:**

- Limit is 0: 3x as x approaches 0
- Limit is 2: (x squared - 1)/(x - 1) as x approaches 1, (x squared - 2x)/(x - 2) as x approaches 2
- Limit does not exist: 1/x as x approaches 0

**Why:** Items 2 and 4 both simplify and evaluate to 2; items 1 and 6 need only substitution. The two failures fail for different reasons: 1/x blows up without settling anywhere, while the absolute-value quotient settles on two DIFFERENT values from the two sides.

Page: https://tryals.app/practice/mathematics-i/limits/sort-each-limit-by-its-value