Limits at Infinity

A **limit at infinity** describes what happens to $f(x)$ when $x$ grows without bound (or decreases without bound). $$\lim_{x \to \infty} f(x) = L$$ means: as $x$ takes larger and larger values ($10, 100, 1000, \ldots$), the values $f(x)$ get as close to $L$ as we like. In the same way, $\lim_{x \to -\infty} f(x) = L$ looks at $x = -10, -100, -1000, \ldots$ When such a finite limit exists, the line $y = L$ is a **horizontal asymptote** of the graph. **The basic building block:** for any positive power $n$, $$\lim_{x \to \pm\infty} \frac{1}{x^n} = 0$$ A fixed number divided by a huge number is tiny. Also $\lim_{x \to \infty} e^{-x} = 0$, because $e^{-x} = \frac{1}{e^x}$. **Rational functions:** compare the **degree** of the numerator ($n$) with the degree of the denominator ($m$): - $n < m$: the limit is $0$ (asymptote $y = 0$) - $n = m$: the limit is the ratio of the **leading coefficients** - $n > m$: there is no finite limit (no horizontal asymptote) **Why it works:** divide every term by the highest power of $x$ in the denominator. Every term of the form $\frac{c}{x^k}$ goes to $0$ and only the leading terms are left. For example, $\frac{2x^2 + 1}{x^2 + 3} = \frac{2 + \frac{1}{x^2}}{1 + \frac{3}{x^2}} \to \frac{2}{1} = 2$.

A Term That Fades Away

Find $\lim_{x \to \infty} \left(3 + \frac{5}{x^2}\right)$.

Try a few large values of x: $x = 10$: $3 + \frac{5}{100} = 3.05$; $x = 100$: $3 + \frac{5}{10000} = 3.0005$ = The values approach $3$

Use the basic rule: $\frac{5}{x^2} = 5 \cdot \frac{1}{x^2}$ and $\frac{1}{x^2} \to 0$ = $\frac{5}{x^2} \to 0$

Add the limits: $3 + 0$ = $3$

Answer: $\lim_{x \to \infty} \left(3 + \frac{5}{x^2}\right) = 3$, so $y = 3$ is a horizontal asymptote.

Equal Degrees: Ratio of Leading Coefficients

Find $\lim_{x \to \infty} \frac{6x^2 - 1}{2x^2 + 5}$.

Compare the degrees: Numerator: degree $2$. Denominator: degree $2$ = Equal degrees

Divide every term by $x^2$: $\frac{6x^2 - 1}{2x^2 + 5} = \frac{6 - \frac{1}{x^2}}{2 + \frac{5}{x^2}}$ = Only the fractions with $x$ in the denominator change

Let x grow without bound: $\frac{1}{x^2} \to 0$ and $\frac{5}{x^2} \to 0$, so the expression approaches $\frac{6 - 0}{2 + 0}$ = $\frac{6}{2} = 3$

Check with a large value: $x = 100$: $\frac{59999}{20005} \approx 2.9992$ = Very close to $3$

Answer: $\lim_{x \to \infty} \frac{6x^2 - 1}{2x^2 + 5} = 3$. The horizontal asymptote is $y = 3$.

Smaller Degree on Top

Find $\lim_{x \to \infty} \frac{3x + 1}{x^2 + 1}$.

Compare the degrees: Numerator: degree $1$. Denominator: degree $2$ = The bottom grows faster

Divide every term by $x^2$: $\frac{\frac{3}{x} + \frac{1}{x^2}}{1 + \frac{1}{x^2}}$ = Numerator $\to 0$, denominator $\to 1$

Find the limit: $\frac{0 + 0}{1 + 0}$ = $0$

Answer: $\lim_{x \to \infty} \frac{3x + 1}{x^2 + 1} = 0$, so $y = 0$ (the $x$-axis) is the horizontal asymptote. At $x = 100$ the value is $\frac{301}{10001} \approx 0.03$.

Going to Negative Infinity

Find $\lim_{x \to -\infty} \frac{4x^3 - x}{2 - 8x^3}$.

Find the leading terms: Numerator: $4x^3$. Denominator: $-8x^3$ (the highest power, even though it is written last) = Both have degree $3$

Divide every term by $x^3$: $\frac{4 - \frac{1}{x^2}}{\frac{2}{x^3} - 8}$ = The small fractions go to $0$ for $x \to -\infty$ too

Take the ratio of leading coefficients: $\frac{4}{-8}$ = $-\frac{1}{2}$

Check with a value: $x = -10$: $\frac{-3990}{8002} \approx -0.4986$ = Close to $-0.5$

Answer: $\lim_{x \to -\infty} \frac{4x^3 - x}{2 - 8x^3} = -\frac{1}{2}$. The horizontal asymptote is $y = -\frac{1}{2}$ (on both ends).

Bigger Degree on Top: No Finite Limit

Describe $\lim_{x \to \infty} \frac{x^3}{x^2 + 1}$.

Compare the degrees: Numerator: degree $3$. Denominator: degree $2$ = The top grows faster

Divide every term by $x^2$: $\frac{x}{1 + \frac{1}{x^2}}$ = The denominator $\to 1$, but the numerator $x$ keeps growing

Test large values: $x = 10$: $\frac{1000}{101} \approx 9.9$; $x = 100$: $\frac{1000000}{10001} \approx 99.99$ = The values grow without bound

Answer: There is no finite limit: $\frac{x^3}{x^2 + 1}$ grows without bound, written $\lim_{x \to \infty} \frac{x^3}{x^2 + 1} = \infty$. The graph has no horizontal asymptote. (Extension beyond this lesson: far to the right, the graph follows the slanted line $y = x$.)

Limits at infinity answer the question "what happens in the long run?": - **Science**: hot coffee cools toward room temperature; the limit of its temperature model is that room temperature - **Business**: the average cost per item falls toward the cost of materials as production grows, because fixed costs are spread thin - **Medicine and chemistry**: concentrations in a mixing tank or in the blood settle at a steady level - **Graphing**: horizontal asymptotes show the end behavior of rational functions at a glance - **Calculus**: improper integrals, series and many models in physics are built on limits at infinity This lesson builds on Introduction to Limits and Limit Notation: there you met the symbol $x \to \infty$, here you learn to compute such limits quickly and reliably.

Mistake: Plugging in infinity: $\frac{\infty}{\infty} = 1$

Why: $\infty$ is not a number, so you cannot substitute it or cancel it. Numerator and denominator can grow at very different speeds.

Correct: Divide by the highest power of $x$ first. For $\frac{6x^2 - 1}{2x^2 + 5}$ the limit is $3$, not $1$.

Mistake: Using the first written coefficient instead of the leading one: $\frac{4x^3 - x}{2 - 8x^3} \to \frac{4}{2} = 2$

Why: The leading coefficient belongs to the term with the highest power, wherever it stands in the expression.

Correct: The leading term of $2 - 8x^3$ is $-8x^3$, so the limit is $\frac{4}{-8} = -\frac{1}{2}$.

Mistake: Swapping the rules: "top degree smaller means no limit"

Why: When the denominator has the bigger degree, it grows much faster, which pushes the fraction toward $0$.

Correct: Smaller degree on top: limit $0$. Bigger degree on top: no finite limit. Equal degrees: ratio of leading coefficients.

Mistake: Believing a graph can never cross its horizontal asymptote

Why: A horizontal asymptote only describes the far left and far right of the graph. In the middle the curve may cross it.

Correct: $y = \frac{3x + 1}{x^2 + 1}$ has the asymptote $y = 0$ and still crosses it at $x = -\frac{1}{3}$.

Cooling Coffee

Newton's law of cooling says a hot drink cools toward the temperature of the room. The long-run temperature is a limit at infinity.

A cup of tea follows $T(t) = 22 + 60e^{-0.05t}$ (degrees Celsius, $t$ in minutes). Since $e^{-0.05t} \to 0$, the tea approaches $22°C$.

Average Cost in a Workshop

Fixed costs (rent, machines) are shared by all items produced. The more items, the smaller each one's share, so the average cost approaches the cost per item.

A bakery with 300 dollars of daily fixed costs and 2 dollars of ingredients per cake has average cost $\frac{300 + 2x}{x}$, which approaches 2 dollars per cake.

Salt Concentration in a Tank

Chemical engineers model how a mixture changes over time. When salty water keeps flowing into a tank, the concentration settles at a steady value.

If salt in a tank grows like $50t$ grams while the volume is $100 + 2t$ litres, the concentration $\frac{50t}{100 + 2t}$ approaches $\frac{50}{2} = 25$ grams per litre.

$\lim_{x \to \infty} f(x) = L$ means $f(x)$ gets as close to $L$ as we like when $x$ grows without bound; $x \to -\infty$ works the same way to the left

Basic facts: $\frac{1}{x^n} \to 0$ for any positive $n$ as $x \to \pm\infty$, and $e^{-x} \to 0$ as $x \to \infty$

To find the limit of a rational function, divide every term by the highest power of $x$ in the denominator

Degree on top smaller: limit $0$. Equal degrees: ratio of leading coefficients. Degree on top bigger: no finite limit

A finite limit $L$ at infinity means $y = L$ is a horizontal asymptote; a graph may still cross it in the middle

Never substitute $\infty$: expressions like $\frac{\infty}{\infty}$ have no fixed value

Q: Can I just plug in a very big number instead of computing the limit?

A: A big number such as $x = 1000$ is a good check and often shows the trend, but it is not a proof. The algebra (dividing by the highest power) tells you the exact value, for example exactly $3$ instead of $2.9992$.

Q: Are the limits at $\infty$ and at $-\infty$ always the same?

A: For rational functions with a finite limit, yes: both ends give the same horizontal asymptote. Other functions can differ. For example, $e^{-x} \to 0$ as $x \to \infty$, but $e^{-x}$ grows without bound as $x \to -\infty$.

Q: Can a graph cross its horizontal asymptote?

A: Yes. The asymptote describes only the end behavior, far to the left or right. The graph of $\frac{3x + 1}{x^2 + 1}$ crosses $y = 0$ at $x = -\frac{1}{3}$ and then approaches it again.

Q: What happens when the numerator's degree is exactly one more than the denominator's?

A: This is an extension beyond this lesson. There is no horizontal asymptote, but the graph approaches a slanted line (an oblique asymptote). For $\frac{x^3}{x^2 + 1}$ that line is $y = x$. You find it with polynomial division.

Exit Presentation

Limits at Infinity

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Pre-Calculus / Limits

Limits at Infinity

Find what a function approaches as x grows without bound, compare degrees of rational functions and connect limits at infinity to horizontal asymptotes.

Advanced30 min

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