Limits, One-Sided Limits, Vertical Asymptotes, Squeeze Theorem, and Applications

The two-sided limit exists if and only if the left-hand limit and the right-hand limit exist and are equal:
\lim{x\to a} f(x) = L \quad\text{iff}\quad \lim{x\to a^-} f(x) = L \text{ and } \lim_{x\to a^+} f(x) = L.
If the left and right limits both go to +\infty or both go to -\infty, then the two-sided limit is that infinite value:
\lim{x\to a} f(x) = +\infty\quad\text{if}\quad \lim{x\to a^-} f(x)=+\infty \text{ and } \lim{x\to a^+} f(x)=+\infty; \lim{x\to a} f(x) = -\infty\quad\text{if}\quad \lim{x\to a^-} f(x)=-\infty \text{ and } \lim{x\to a^+} f(x)=-\infty.
If the left and right limits are not equal, the two-sided limit does not exist:
\lim_{x\to a} f(x) \;\text{does not exist (DNE)}.
Strategy: when asked for a limit at a point that looks like a vertical asymptote, first analyze the two-sided limit by examining the one-sided limits.

For a one-sided limit (plus or minus), the sign (positive infinity vs negative infinity) is determined by evaluating the function at a nearby point on the appropriate side of the asymptote.
Reason:
- Near a vertical asymptote, the function must tend to either +∞ or -∞. The nearby value tells you which sign the approach has, because the function cannot jump to a finite value suddenly.
Practical method:
- Plug in a nearby value from the left (e.g., a − ε) to determine the left-hand behavior.
- Plug in a nearby value from the right (e.g., a + ε) to determine the right-hand behavior.
Example 1: near x = 1 for f(x) = -4/(1 - x)
- Left limit (x → 1⁻): take x = 0.9 → 1 - x = 0.1 > 0, so f(0.9) = -4/0.1 = -40 ⇒ goes to -∞
- Right limit (x → 1⁺): take x = 1.1 → 1 - x = -0.1 < 0, so f(1.1) = -4/(-0.1) = 40 ⇒ goes to +∞
- Conclusion: two-sided limit does not exist because left and right limits have opposite infinite signs.
Example 2: near x = π/2 for f(x) = x^3/cos(x)
- Left (x → π/2⁻): cos(x) > 0, x^3 > 0 → f → +∞
- Right (x → π/2⁺): cos(x) < 0, x^3 > 0 → f → -∞
- Conclusion: two-sided limit does not exist.
Note on signs without calculators:
- If you rewrite a function to a simpler form, you can deduce sign changes from the known signs of factors on each side of the asymptote (e.g., sign of numerator vs sign of denominator near the asymptote).

When an absolute value is present, take care to use its definition first:
- For example, around x = 5,
  |x-5| = \begin{cases} x-5, & x \ge 5, \ -(x-5) = 5-x, & x < 5. \end{cases}
Then evaluate left-hand and right-hand limits by substituting from the appropriate branch.
In the transcript, this approach was used to evaluate limits where the absolute value sign changes the expression on each side.

A typical strategy is to rewrite the function so that the factor causing the asymptote is explicit.
- Example: f(x) = \frac{x^3}{\cos x}.
- Vertical asymptotes occur where the denominator is zero, i.e., where \cos x = 0.
- In the interval [0, π], cos x = 0 at x = π/2, so x = π/2 is a vertical asymptote.
- Left-hand limit near π/2: \lim{x\to(π/2)^-} f(x) = +∞; Right-hand limit: \lim{x\to(π/2)^+} f(x) = -∞.
A quick geometric intuition can help: near a vertical asymptote, the function shoots off to ±∞ and cannot jump to a finite value as x approaches a from either side.

Secant line between two points on a graph of f: points (x1, f(x1)) and (x2, f(x2)).
- Slope of the secant line (average rate of change):
  m{sec} = \frac{f(x2) - f(x1)}{x2 - x_1}.
If the two points are h units apart: x1 = x, x2 = x + h, then
m_{sec}(x,h) = \frac{f(x+h) - f(x)}{h}.
As h → 0, the secant slope tends to the tangent slope at x:
\lim_{h\to 0} \frac{f(x+h) - f(x)}{h} = f'(x),
which is the slope of the tangent line at x and the instantaneous rate of change.
Interpretations in physics:
- If f represents a position function s(t), then the derivative f'(t) is the velocity.
- The average velocity on [t1, t2] is \frac{s(t2) - s(t1)}{t2 - t1}.
- The instantaneous velocity is the limit of the average velocity as the time interval shrinks to zero, i.e., the derivative at that time.

Idea: if three functions satisfy f(x) ≤ g(x) ≤ h(x) near a (excluding a), and
\lim{x\to a} f(x) = \lim{x\to a} h(x) = L,
then
\lim_{x\to a} g(x) = L.
This is a standard, powerful tool for proving limits when g is trapped between two functions with the same limit.
Simple illustration (concrete example):
- Let a = 4. Define
  f(x) = -(x-4)^2, \quad g(x) = 0, \quad h(x) = (x-4)^2.
- For x near 4, indeed f(x) ≤ g(x) ≤ h(x).
- As x → 4, \lim{x\to 4} f(x) = \lim{x\to 4} h(x) = 0.
- By the squeeze theorem, \lim_{x\to 4} g(x) = 0.
A three-function setup: if f ≤ g ≤ h and both f and h approach the same limit L, then g must also approach L.
A classic challenging problem (use of squeeze): determine limits involving cosines that oscillate. The standard approach is to bound a cosine expression and apply the squeeze.

Consider the limit
\lim_{x\to 0} x^2\cos x.
Use the bound for cosine: -1 \le \cos x \le 1, hence
-x^2 \le x^2\cos x \le x^2.
Since \lim{x\to 0} -x^2 = \lim{x\to 0} x^2 = 0, by the squeeze theorem,
\lim_{x\to 0} x^2\cos x = 0.
Note: a related common variant is \lim_{x\to 0} x\cos\left(\frac{1}{x}\right) which also tends to 0 by the same logic (bounded cosine times a vanishing x).

Infinite limits (divergence to ±∞) are not real numbers, but are still meaningful limit statements:
- If both one-sided limits go to +∞, declare the two-sided limit as +∞.
- If one side goes to +∞ and the other to -∞, the two-sided limit does not exist (DNE).
If a limit is not a real number and cannot be assigned a specific infinite value that matches both sides, we say DNE.

Always start with the nature of the point a: is it a vertical asymptote? If yes, examine one-sided limits to determine signs.
Use nearby points to infer the sign of the limit when dealing with a vertical asymptote.
When absolute value is present, prefer the piecewise definition to simplify the limit from each side.
Rewrite the function to expose the problematic factor (e.g., denominator approaching zero) to identify vertical asymptotes and apply one-sided limits.
Leverage the secant/tangent discussion to connect limits with the idea of instantaneous rate of change and velocity in physical problems.
Apply the squeeze theorem where you can bound a function between two others that share the same limit, especially with oscillatory components like cosine or sine.

Limits, One-Sided Limits, Vertical Asymptotes, and the Squeeze Theorem – Study Notes