Limit at infinity, equal degrees — the ratio of leading coefficients is the whole answer
Evaluate
$$\lim_{x \to \infty} \frac{x^3 - 2x^2 + x - 3}{3x^3 - 2x + 1}$$Answer: $\dfrac{1}{3}$
Look at the degrees first, the way you did in Q4:
They're the same. That changes everything.
When top and bottom grow at the same rate, the fraction settles down to a single finite number — the ratio of how fast they grow. Like two trains at the same speed: the gap between them stays constant. For polynomials, the "growth rate" of a degree-3 polynomial is set by its leading coefficient, so the limit is just leading coefficient of top ÷ leading coefficient of bottom.
Here: $\frac{1}{3}$.
For $\displaystyle\lim_{x \to \infty} \frac{a_n x^n + \text{lower}}{b_n x^n + \text{lower}}$ with $a_n, b_n \neq 0$ and the same degree $n$ on top and bottom:
$$\lim_{x \to \infty} \frac{a_n x^n + \cdots}{b_n x^n + \cdots} = \frac{a_n}{b_n}$$Why? Factor out the highest power on both sides:
$$\frac{x^n\!\left(a_n + \frac{a_{n-1}}{x} + \cdots\right)}{x^n\!\left(b_n + \frac{b_{n-1}}{x} + \cdots\right)} = \frac{a_n + \frac{a_{n-1}}{x} + \cdots}{b_n + \frac{b_{n-1}}{x} + \cdots}$$As $x \to \infty$, every $\frac{1}{x}$ term vanishes, leaving $\frac{a_n}{b_n}$.
Step 1 — Identify the leading terms.
Step 2 — Factor $x^3$ out of both.
$$\frac{x^3 - 2x^2 + x - 3}{3x^3 - 2x + 1} = \frac{x^3\!\left(1 - \frac{2}{x} + \frac{1}{x^2} - \frac{3}{x^3}\right)}{x^3\!\left(3 - \frac{2}{x^2} + \frac{1}{x^3}\right)}$$Step 3 — Cancel $x^3$.
$$= \frac{1 - \frac{2}{x} + \frac{1}{x^2} - \frac{3}{x^3}}{3 - \frac{2}{x^2} + \frac{1}{x^3}}$$Step 4 — Take the limit. All the $\frac{1}{x^k}$ terms go to 0:
$$= \frac{1 - 0 + 0 - 0}{3 - 0 + 0} = \frac{1}{3}$$Answer: $\dfrac{1}{3}$.
| Choice | Verdict | Why |
|---|---|---|
| $\infty$ | Wrong | That's the Q4 trap — top degree strictly greater than bottom degree. Here they're equal, so the limit is a finite number, not infinity. |
| 3 | Wrong | You got the ratio upside down. The leading coefficient on top is $1$, the one on the bottom is $3$, so the ratio is $1/3$, not $3/1$. (The fraction 3 alone is what you'd get if you had $\frac{3x^3}{x^3}$.) |
| 0 | Wrong | That's the case where the bottom grows faster (degree strictly bigger). Here they're equal, so the limit is nonzero and finite. |
| $\dfrac{1}{3}$ | Correct | Equal degrees → ratio of leading coefficients = $\frac{1}{3}$. |
| Top degree vs bottom degree | Limit | What it means |
|---|---|---|
| Top < bottom | $0$ | Bottom grows faster → fraction shrinks to 0 |
| Top = bottom | $\dfrac{a_n}{b_n}$ (ratio of leading coefficients) | Both grow at same rate → settle to a constant |
| Top > bottom | $\pm\infty$ | Top grows faster → blows up |
Q4 = middle row of the previous lesson (top wins → $\infty$). Q5 = middle row of this lesson (equal → ratio of leading coefficients). Once you can recognize which case you're in, the answer is mechanical.
Both degree 2. Equal → ratio of leading coefficients.
$\frac{4}{2} = 2$.
Answer: $2$.
Both degree 4. Equal → ratio of leading coefficients.
$\frac{6}{2} = 3$.
Answer: $3$.
Top degree 2, bottom degree 3. Bottom wins → $0$.
(Don't fall for the trap of "the leading coefficient ratio is $9/3 = 3$" — that's only valid when degrees are equal.)
Answer: $0$.