Limit at infinity, polynomial-over-polynomial, top degree wins
Evaluate
$$\lim_{x \to \infty} \frac{3x^2 - 2x + 5}{x}$$Answer: $\infty$
"$x \to \infty$" means "$x$ grows without bound." The question becomes: which part of this fraction blows up faster, the top or the bottom?
Look at the degrees:
When you divide something that grows like $x^2$ by something that grows like $x$, you get something that grows like $x$. So the limit runs off to $\infty$.
For a rational function $\dfrac{p(x)}{q(x)}$ where $p, q$ are polynomials, only the highest-degree terms matter as $x \to \infty$. The result depends on the degree comparison:
| Case | Limit | Why |
|---|---|---|
| $\deg p < \deg q$ | $0$ | Bottom grows faster → fraction shrinks to 0 |
| $\deg p = \deg q$ | $\dfrac{a_n}{b_n}$ | Ratio of leading coefficients |
| $\deg p > \deg q$ | $\pm\infty$ | Top grows faster → fraction blows up |
Q4 is the third row: degree 2 > degree 1, so the limit is $\pm\infty$. Since the leading coefficient $3$ is positive, it's $+\infty$.
Step 1 — Compare degrees. Top is degree 2, bottom is degree 1. Top wins, so the limit is $\pm\infty$.
Step 2 — Verify by dividing every term in the top by the bottom.
$$\frac{3x^2 - 2x + 5}{x} = \frac{3x^2}{x} - \frac{2x}{x} + \frac{5}{x} = 3x - 2 + \frac{5}{x}$$Step 3 — Take the limit term-by-term.
$$\lim_{x \to \infty} \left(3x - 2 + \frac{5}{x}\right) = \underbrace{3x}_{\to\, \infty} \;-\; \underbrace{2}_{\to\, 2} \;+\; \underbrace{\frac{5}{x}}_{\to\, 0}$$The $-2$ and the $\frac{5}{x}$ become irrelevant. The $3x$ term goes to $\infty$.
Answer: $\infty$.
| Choice | Verdict | Why |
|---|---|---|
| 3 | Wrong | That's the answer when degrees are equal and you take the ratio of leading coefficients (Q5). Here, the top is degree 2, the bottom is degree 1 — the answer can't be a single number. |
| 0 | Wrong | That's the answer when the bottom degree is bigger than the top (Q2-style). Here, the top wins. |
| DNE | Wrong | The limit doesn't "not exist" in the pathological sense — it just grows without bound. In calculus, we do say the limit is $\infty$; DNE is reserved for limits that approach different finite values from each side (jump discontinuities, Q5/Q9 of the main pretest). |
| $\infty$ | Correct | Top degree (2) > bottom degree (1) → top wins → $\infty$. |
Top degree 3, bottom degree 2. Top wins. The expression behaves like $\frac{5x^3}{x^2} = 5x \to \infty$.
Answer: $\infty$.
Top and bottom both degree 2. Equal degrees → ratio of leading coefficients.
$\frac{7x^2}{2x^2} = \frac{7}{2}$.
Answer: $\frac{7}{2}$.
Top degree 1, bottom degree 3. Bottom wins → fraction shrinks to 0.
Answer: $0$.