Limit Laws Using Graphs — Detailed Study Notes

Two functions were supplied graphically: f(x) and g(x).
- Exact algebraic formulas were NOT given; all numerical limits are read directly from the plotted y-values.
- Points of interest: $x = 0,\;1,\;2$ .
Task: Evaluate six limits (labeled a–f) using limit laws.
Requirement: State “does not exist” (DNE) whenever the two-sided limit fails, and explain why.

Sum/Difference Law
$\lim{x\to a}\,[f(x) \pm g(x)] = \lim{x\to a}f(x) \pm \lim_{x\to a}g(x)$ (if both one-sided limits exist)
Product Law
$\lim{x\to a}[f(x)\,g(x)] = \Bigl(\lim{x\to a}f(x)\Bigr)\Bigl(\lim_{x\to a}g(x)\Bigr)$
Quotient Law
$\lim{x\to a}\frac{f(x)}{g(x)} = \frac{\lim{x\to a}f(x)}{\lim_{x\to a}g(x)}$ provided the denominator limit exists and is non-zero.
Power / Polynomial Law
$\lim_{x\to a}x^n = a^n$ (continuous everywhere).
Constant Law
$\lim_{x\to a}c = c$
Root Law
$\lim{x\to a}\sqrt[n]{\,h(x)\,} = \sqrt[n]{\,\lim{x\to a}h(x)\,}$ when the inside limit is positive (or otherwise lies in the domain of the root).
Continuity Shortcut
• If a function is continuous at $x=a$ , the limit equals the function value: $\lim_{x\to a}f(x)=f(a)$ .

Target $x$	$f(x)$ (approach)	$g(x)$ (approach)	Notes
$0$	$0$	$\approx1.4$	f crosses the origin; g sits slightly above 1.
$1$	$1$	LEFT (≈2) vs. RIGHT (≈0) → mismatch → limit DNE	Jump discontinuity in g.
$2$	$2$	$0$	f is continuous there; g touches the x-axis.

(The instructor constantly references these values while applying the laws.)

Split via Sum Law: $\lim{x\to2}f(x)+\lim{x\to2}g(x).$
From the graph: $\lim{x\to2}f(x)=2,$ $\lim{x\to2}g(x)=0.$
Result: $2+0 = 2.$
Comment: Both functions are continuous at $x=2$ , so direct substitution was valid.

Apply Sum Law again, but need each individual limit.
• $\lim{x\to1}f(x)=1$ (f continuous). • $\lim{x\to1}g(x)$ does not exist – left-hand limit $\approx2$ , right-hand $\approx0$ .
Because one addend’s limit fails, the entire sum’s limit does not exist (DNE).
Explanation provided: “two sides don’t match up.”

Quotient Law requires both numerator & denominator limits.
• Numerator: $\lim{x\to1}f(x)=1.$ • Denominator: $\lim{x\to1}g(x)$ DNE (same jump issue as part b).
Because the denominator limit fails, the quotient’s limit does not exist.
Instructor’s phrase: “This limit isn’t going to exist either because the limit in the denominator doesn’t exist.”

View it as a product $x^3 \cdot f(x)$ .
Product Law ⇒ $\bigl(\lim{x\to2}x^3\bigr)\bigl(\lim{x\to2}f(x)\bigr).$
1. Power Law: $\lim_{x\to2}x^3 = 2^3 = 8.$
2. From the graph: $\lim_{x\to2}f(x)=2.$
Multiply: $8\times2 = 16.$
Note: Demonstrates combining “basic” polynomial limit with a graph-read limit.

Root Law lets us pull the limit inside:
$= \sqrt{\,\lim_{x\to1}[3+f(x)]\,}.$
Sum Law inside the radical:
$= \sqrt{\,\lim{x\to1}3 + \lim{x\to1}f(x)\,}.$
Constant Law: $\lim_{x\to1}3 = 3.$
From graph: $\lim_{x\to1}f(x)=1.$
Combine inside: $3 + 1 = 4.$
Evaluate root: $\sqrt{4} = 2.$

Valid because $3+f(x)$ stays positive near $x=1$ , preserving domain of the square root.

Graphical intuition is often faster than algebra for limit evaluation—provided discontinuities are recognized.
Sum, product, and quotient laws all require the individual component limits to exist first; one bad component kills the combined limit.
The Power Law and Constant Law are examples of continuity of basic elementary functions.
Root Law extends continuity to radicals, but only where the interior expression remains within the radical’s domain.
When using the quotient law, always check that the denominator limit is non-zero as well as existing; here, the issue was total non-existence.
Mixing multiple laws (as in parts e & f) shows how composite expressions can be broken apart systematically.

Continuous at a point? → Evaluate directly.
Jump in one side vs. the other? → Limit fails.
Product includes a factor whose limit is 0? → Overall limit is 0 (provided the other factor is finite).
Quotient denominator limit DNE or =0? → Combined limit DNE (or ±∞ if one-sided analysis applies).
Root of a limit vs. limit of a root? → Interchangeable if the inside stays in the domain.

These notes encapsulate the logic, laws, and numeric conclusions the instructor walked through in the video.