Notes on Limits to Infinity, Horizontal Asymptotes, and Exponents/Logs (1.14–1.15)

End behavior and horizontal asymptotes

A horizontal asymptote y = l is defined by the behavior of f(x) as x grows without bound in either direction:
- \lim{x\to\infty} f(x) = l or \lim{x\to-\infty} f(x) = l.
- There can be more than one horizontal asymptote for a given function.
Practical takeaway: a horizontal asymptote describes where the graph levels off as x becomes very large or very small.
Example from the transcript (intuitive):
- With a function f of x, filling a table with calculator values shows something like f(1) ≈ 0.5 and f(-1) ≈ 0.5, mirroring on both sides.
- As x grows large in magnitude, the y-values approach 2:
- \lim{x\to\infty} f(x) = 2, \quad \lim{x\to-\infty} f(x) = 2.
- Graph would show a horizontal asymptote at y = 2.
Connection to limits: whenever a limit to infinity yields a finite value, that finite value is the horizontal asymptote.
Quick note from the lecture: the language often uses "as x approaches infinity" to describe approaching behavior, which ties directly to limits.

Horizontal asymptote definition and intuition (/calculus view)

A horizontal asymptote is the line y = l that the graph approaches as x becomes very large or very small.
The two-way condition is enough to declare an asymptote: if either
- \lim{x\to\infty} f(x) = l or \lim{x\to-\infty} f(x) = l
  is true, then y = l is a horizontal asymptote.
There can be more than one horizontal asymptote (one as x → ∞ and another as x → −∞).

Rational functions and limits to infinity (leading-term rule)

When taking limits to infinity for rational functions (quotients of polynomials), compare degrees:
- If the degree of the numerator is less than the degree of the denominator, the limit is 0 (the function is bottom-heavy):
- \deg(\text{numerator}) < \deg(\text{denominator}) \Rightarrow \lim_{x\to\infty} \frac{N(x)}{D(x)} = 0.
- If the degrees are equal, the limit equals the ratio of the leading coefficients:
- \deg(\text{numerator}) = \deg(\text{denominator}) \Rightarrow \lim_{x\to\infty} \frac{N(x)}{D(x)} = \frac{a}{b},
  where a and b are the leading coefficients of the numerator and denominator, respectively.
- If the degree of the numerator is greater than the degree of the denominator (top-heavy), the limit is ±∞ (does not exist as a finite number):
- \deg(\text{numerator}) > \deg(\text{denominator}) \Rightarrow \lim_{x\to\infty} \frac{N(x)}{D(x)} = \pm\infty.
Tip: Infinity is not a number; you never plug in x = ∞. Instead, you examine the growth rates of the numerator and denominator to determine the limit.
The video notes that you may see one or more horizontal asymptotes depending on how the function behaves at ±∞.
Application idea: identify which term dominates as x becomes large (leading terms) and simplify accordingly.

A concrete example: square-root expression and absolute value considerations

Consider the limit related to a square-root expression:
- \lim_{x\to\infty} \sqrt{\frac{2x^2}{3x}}.
Important subtlety: (\sqrt{x^2}) is not simply x; it equals |x| (the nonnegative value).
- Graphically, (\sqrt{x^2}) traces a V-shaped absolute value function near the origin.
Algebraic manipulation (as described in the video):
- Use the property that for positive x, sqrt(a b) = sqrt(a) sqrt(b) and separate inside the radical:
- \sqrt{\frac{2x^2}{3x}} = \frac{\sqrt{2x^2}}{\sqrt{3x}} = \frac{\sqrt{2}\,|x|}{\sqrt{3}\,\sqrt{x}}.
- For x → ∞, |x| = x, so this becomes
- \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{x}{\sqrt{x}} = \frac{\sqrt{2}}{\sqrt{3}} \sqrt{x} \to \infty.
The video then discusses choosing a branch for the absolute value based on the direction of the limit (x → ∞ implies the positive branch). The takeaway is that the limit grows without bound in this case, not converging to a finite value.
Note: The instructor also attempted a further simplification to a finite constant and claimed a horizontal asymptote at that constant; the mathematically consistent result here is that the limit diverges to +∞. The discrepancy is a teaching moment about careful algebra with absolute values inside radicals.

Finite limits (limits to a number) and pattern-finding with tables

When a limit is finite (i.e., not ∞ or −∞), you cannot rely on "leading-term" comparisons of degrees unless you are dealing with a rational function.
The video emphasizes using a table/pattern approach to understand how the function behaves as x approaches a finite value (e.g., near a vertical asymptote where the expression might blow up).
Example pattern approach discussed:
- Consider a limit like \lim_{x\to 2^+} \frac{1}{x-2}.
- Plug in values approaching 2 from the right: x = 2.1, 2.01, 2.001, …
- Then the inner “denominator” behaves as x−2 → 0^+ (positive small), so the whole expression grows without bound to +∞.
- In the same problem, approaching from the left yields negative values approaching −∞.
A second related example:
- \lim_{x\to 4^-} \frac{1}{x-4}.
- Choose x = 3.9, 3.99, 3.999, …; here x−4 → 0^− (negative small), so 1/(x−4) → −∞.
A standard technique to handle limits as x → −∞ is to transform the limit to a positive-infinity form by substituting x → −t, so that as x → −∞, t → ∞, and proceed with the positive-infinity analysis.
The key idea is to extract the dominant behavior (the pieces that grow without bound) and ignore the smaller, non-dominant parts when x is large or close to a problematic finite point.

Exponents and logarithms (recap of inverses and asymptotes)

Exponential function y = e^x:
- Graphically, exponential growth has a typical shape where f(0) = e^0 = 1.
- Anchor points (from the lecture):
- At x = 0, f(x) = 1; at x = 1, f(x) = e; at x = −1, f(x) = e^{−1} = 1/e.
- As x → ∞, e^x → ∞; as x → −∞, e^x → 0^+.
- The exponential graph has no horizontal asymptote on the right; as x → −∞ it approaches 0 but never touches it (0 is a horizontal asymptote on the left for the exponential function when viewed at y-values).
Natural logarithm y = \ln x:
- Domain is x > 0; vertical asymptote at x = 0^+ (ln x → −∞ as x → 0^+).
- Anchor points mentioned: ln(1) = 0 and ln(e) = 1; ln is undefined for x ≤ 0.
- The graph increases without bound as x → ∞: ln x → ∞.
Inverses and the x-y swap:
- Exponentials and logarithms are inverse functions of each other: swapping x and y in y = e^x yields the graph of y = \ln x.
- Because they are inverses, their asymptotes switch roles:
- The horizontal asymptote for the exponential (as x → −∞, y → 0) corresponds to a vertical asymptote for the log (at x = 0).
- Anchor points for the inverse pair (derived from the inverses):
- For e^x: (0,1) and (1,e).
- For ln x: (1,0) and (e,1).
Quick limit reminders for these functions:
- \lim{x\to\infty} e^{x} = \infty, \quad \lim{x\to-\infty} e^{x} = 0.
- \lim{x\to 0^+} \ln x = -\infty, \quad \lim{x\to\infty} \ln x = \infty.
- The relation between the two is an inverse: if y = e^x, then x = \ln y.
Practical takeaway: remembering anchor points and the inverse relationship helps sketch and reason about limits and asymptotes quickly.

Quick practice: limits to infinity with exponentials/logs

Example patterns to recall:
- Exponential growth: \lim_{x\to\infty} e^{x} = \infty (no finite horizontal asymptote on the right).
- Exponential decay: \lim_{x\to-\infty} e^{x} = 0.
- Logarithm growth: \lim{x\to\infty} \ln x = \infty and the logarithm blows up near the left boundary: \lim{x\to 0^+} \ln x = -\infty.
The video emphasizes pausing to review precalculus concepts if needed (exponents/logs as inverses, anchor points, domain/range implications).

Worked finite-limit examples (patterns without infinity)

Example pattern idea: finite limit near a vertical asymptote cannot be determined by naive substitution; you must examine behavior from left and right or use a table/pattern.
Classic example: 1/(x−2) around x = 2
- Right-hand limit:
- Choose x-values approaching 2 from the right: 2.1, 2.01, 2.001, …
- Evaluate 1/(x−2): 1/0.1 = 10, 1/0.01 = 100, 1/0.001 = 1000, … → +∞.
- Left-hand limit:
- Values approaching from the left would give negative large numbers, indicating −∞.
Another example: 1/(x−4) as x → 4^−
- Choose x = 3.9, 3.99, 3.999, …
- Compute 1/(x−4): 1/(−0.1) = −10, 1/(−0.01) = −100, 1/(−0.001) = −1000, … → −∞.
A common technique for limits to −∞:
- If you need to handle a limit to −∞ as x → −∞, you can substitute x = −t, turning it into a limit to +∞ in t and proceed analogously.

Summary and study tips

Always identify the type of limit: finite value, ∞, or −∞, before choosing a method.
For limits to infinity of rational functions, compare degrees to determine the outcome (0, finite ratio, or ±∞).
For non-rational or mixed functions, isolate the dominant term(s) that determine growth (leading behavior) and simplify accordingly.
When dealing with exponentials and logs, use the inverse relationship to switch between graphs and limits when helpful, and remember the basic anchor points:
- Exponential: e^0 = 1, e^1 = e, e^{−1} = 1/e; limit behavior as x → ±∞.
- Log: domain x > 0, ln(1) = 0, ln(e) = 1; vertical asymptote at x = 0; inverse relationship with e^x.
Practice with the provided Delta Math assignments and progress checks; use Schoology resources for answer keys and extended problems.

Note on the video flow

The instructor indicated a split into two parts and recommended pausing to work problems, then resuming to see the solution flow.
They stressed showing full work for limit reasoning (especially for limits to infinity) and the importance of not skipping steps when grading.
They also reminded that some problems (finite limits) require a different strategy than simple leading-term analysis used for infinity limits.
The takeaway is a solid foundation in horizontal asymptotes, end behavior of rational functions, and the basics of exponents and logarithms as precalculus and calculus foundations.
If you want, I can convert these notes into a condensed study guide with fewer or more worked examples, or add additional practice problems similar to the ones described here.