# Limits and continuity

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/limits-and-continuity

## Understanding Limits

A **limit** captures what value $f(x)$ approaches as $x$ gets arbitrarily close to a point $a$.

$$\lim_{x \to a} f(x) = L \;\Longleftrightarrow\; \forall\,\varepsilon>0,\;\exists\,\delta>0 :\; 0<|x-a|<\delta \implies |f(x)-L|<\varepsilon$$

| Limit Type | Example | Meaning | Gateway |
| --- | --- | --- | --- |
| **Finite** | $\lim_{x\to 3}(2x+1)=7$ | Value at point | Direct eval |
| **Infinite** | $\lim_{x\to 0^{+}} \frac{1}{x} = +\infty$ | Vertical asymptote | Divergence |
| **At infinity** | $\lim_{x\to\infty} \frac{1}{x} = 0$ | Horizontal asymptote | End behavior |

**Indeterminate forms** like $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ require algebraic manipulation (factoring, rationalising) or **L'Hôpital's rule** before evaluation.

## Continuity and Pitfalls

A function is **continuous** at $a$ when three conditions hold simultaneously:

1. $f(a)$ exists.
2. $\lim_{x\to a} f(x)$ exists.
3. $\lim_{x\to a} f(x) = f(a)$.

The **Intermediate Value Theorem** states that a function continuous on $[a,b]$ is guaranteed to attain every value between $f(a)$ and $f(b)$.

> **Common pitfall:** $\lim_{x\to a} f(x)$ is about the journey, not the destination. The limit can exist where $f(a)$ is undefined, and can differ from $f(a)$ where defined. **Continuity** is the exact promise that they agree.

## Practice questions

6 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort these functions based on the type of discontinuity.

**Answer:**

- Removable: $f(x) = (x^{2})/x$, $f(x) = \sin(x)/x$
- Non-Removable: $f(x) = 1/x$, $f(x) = |x|/x$, $f(x) = 1/x^{2}$

**Why:** Removable discontinuities can be filled by redefining the point.

Page: https://tryals.app/practice/physics-i/limits-and-continuity/sort-these-functions-based-on-the-type-of-discontinuity

### 2. A function satisfies $f(2) = 5$, but $\lim_{x\to 2} f(x) = 3$. What does the graph look like near $x = 2$?

A. The curve heads toward height 5, with a hole at height 3
B. The curve jumps from 3 up to 5 as $x$ crosses 2
C. The curve heads toward 3, with a lone dot at 5
D. Impossible, the limit must equal the value

**Answer:** C. The curve heads toward 3, with a lone dot at 5

**Why:** This is a removable discontinuity: the curve funnels toward 3, but someone stuck the actual point at 5. Continuity is precisely the demand that these two, the heading and the landing, agree.

Page: https://tryals.app/practice/physics-i/limits-and-continuity/a-function-satisfies-f-2-5-but-lim-x-2-f-x-3-what-does-the

### 3. A function can have a limit of 5 at $x = 2$ while $f(2) = 9$, or while $f(2)$ does not exist at all. What does that show about what a limit measures?

A. That the function must be discontinuous whenever a limit exists
B. That limits are approximations to the function value and are usually close to it
C. That it describes what the function APPROACHES, not its value
D. That the limit and the value agree except at finitely many points

**Answer:** C. That it describes what the function APPROACHES, not its value

**Why:** Deliberately ignoring the point is the whole design. Without it the derivative could not be defined, because the difference quotient is undefined precisely where the derivative is being taken.

Page: https://tryals.app/practice/physics-i/limits-and-continuity/a-function-can-have-a-limit-of-5-at-x-2-while-f-2-9-or-while

### 4. Order the steps in an epsilon-delta proof that $lim_{x\to a} f(x)$ = L.

**Answer:**

1. Let $\varepsilon$ > 0 be given (arbitrarily small)
2. Express |$f(x)$ − L| in terms of |x − a|
3. Choose $\delta$ > 0 in terms of $\varepsilon$ to control |$f(x)$ − L|
4. Verify: |x − a| < $\delta$ implies |$f(x)$ − L| < $\varepsilon$

**Why:** The epsilon-delta definition formalises the intuitive notion of a limit.

Page: https://tryals.app/practice/physics-i/limits-and-continuity/order-the-steps-in-an-epsilon-delta-proof-that-lim-x-a-f-x-l

### 5. Sort each function by what happens at the marked point.

**Answer:**

- Continuous there: $|x|$ at $x = 0$
- Removable hole: $\frac{x^{2}-4}{x-2}$ at $x = 2$
- Jump: Postage cost as weight crosses a price step
- Vertical asymptote: $1/x$ at $x = 0$

**Why:** The three ways continuity fails: a hole (limit exists, point missing), a jump (one-sided limits disagree), a blow-up (no finite limit). The corner of $|x|$ breaks none of these, corners are a problem for **derivatives**, not for continuity.

Page: https://tryals.app/practice/physics-i/limits-and-continuity/sort-each-function-by-what-happens-at-the-marked-point

### 6. Numerically explore $\dfrac{\sin x}{x}$ as $x \to 0$ (radians): compute it mentally for small $x$, then set the slider to the limit.

**Answer:** 1 (within ±0.1)

**Why:** The ratio squeezes to exactly 1: for small angles, $\sin x \approx x$. This limit is the seed of all trigonometric calculus, the derivative of sine at 0, and the reason pendulum formulas work for small swings.

Page: https://tryals.app/practice/physics-i/limits-and-continuity/numerically-explore-sin-x-x-as-x-0-radians-compute-it