🎯⭐ INTERACTIVE LESSON

Direct Substitution Method

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Direct Substitution Method - Complete Interactive Lesson

Part 1: The Idea of Direct Substitution

🎯 Direct Substitution — The Idea

Part 1 of 4 — When you can "just plug in"

Topics in This Part

Section
The Big Idea
🔑 The Continuity Connection
Easy Examples

🔑 Why this matters: Direct substitution is the first thing you try on every limit problem. Many AP limits are solved in one line.

💡 The Big Idea

Direct substitution: to find $\lim_{x \to a} f(x)$ , simply compute $f(a)$ .

If $f$ is continuous at $a$ , the answer is just $f(a)$ — no fancy work needed.

$\lim_{x \to a} f(x) = f(a) \quad \text{whenever $f$ is continuous at $a$.}$

💡 This is the definition of continuity: limit equals value.

🔑 The Continuity Connection

Direct substitution works because of a chain of facts:

Polynomials are continuous everywhere.
Rational functions are continuous everywhere their denominator is nonzero.
$\sin x$ , $\cos x$ , $e^x$ are continuous everywhere.
$\ln x$ is continuous on .

📝 Easy Examples

Limit	Substitution	Answer
$\lim_{x \to 4} (3x^2 - 2x + 1)$

Concept Check 🎯

Substitute 🧮

Compute each limit by direct substitution:

1) $\lim_{x \to 2} (x^3 - x) = ?$

Part 2: When Direct Substitution Works

✅ When Direct Substitution Works

Part 2 of 4 — A continuity catalog

Topics in This Part

Section
Polynomials
Rational Functions
Trig, Exp, Log, Roots

🔑 Why this matters: Knowing the "continuity catalog" lets you spot a one-line problem at a glance.

📐 Polynomials

A polynomial $p(x) = a_n x^n + \cdots + a_0$ is continuous .

Part 3: Limit Laws

📜 Limit Laws — The Algebra of Substitution

Part 3 of 4 — Why "just plug in" works

Topics in This Part

Section
🔑 The Five Core Limit Laws
Composition of Continuous Functions
Examples

🔑 Why this matters: Limit laws are what justify direct substitution and let you break complex limits into pieces.

🔑 The Five Core Limit Laws

If $\lim_{x \to a} f(x) = L$ and (both exist), and is a constant:

Part 4: When Direct Substitution Fails

🚦 When Direct Substitution Fails

Part 4 of 4 — Recognize the trigger and switch tools

Topics in This Part

Section
The Indeterminate Form Trigger
What to Do Next (a Roadmap)
Substitution vs. Indeterminate vs. DNE

🔑 Why this matters: When substitution gives $0/0$ or another indeterminate form, that's a signal — not an answer. You need a different technique.

⚠️ The Indeterminate Form Trigger

Plugging in $x = a$ can yield:

Result	What it means

\sqrt{x}

is continuous on

x \ge 0

Sums, products, quotients (denominator

\ne 0

), and compositions of continuous functions are continuous.

💡 Strategy: before trying anything fancy, try plugging in. If it gives a clean number, you're done.

lim_{x \to 0} (sin x + 5) = ?

3) $\lim_{x \to 4} \dfrac{x + 1}{x - 1} = ?$ (decimal or fraction)

So: $\lim_{x \to a} p(x) = p(a) \quad \text{for any real } a.$

💡 Always try substitution first for polynomials. It will work.

÷ Rational Functions

A rational function $r(x) = \dfrac{p(x)}{q(x)}$ is continuous wherever $q(x) \ne 0$ .

Situation	Result
$q(a) \ne 0$	Direct substitution gives the limit immediately
$q(a) = 0$ but

Worked example: $\lim_{x \to 5} \dfrac{x^2 - 1}{x - 2} = \dfrac{24}{3} = 8$ . ✅

🌀 Trig, Exp, Log, Roots

Function	Continuous on
$\sin x$ , $\cos x$	All of $\mathbb{R}$
$e^x$	All of $\mathbb{R}$
$\ln x$	$x > 0$
$\sqrt{x}$	$x \ge 0$
$\tan x$	All $x$ except odd multiples of $\pi/2$
$\sec x$ , $\csc x$ , $\cot x$	Wherever denominators don't vanish

⚠️ For $\tan x$ at $\pi/2$ , $\sqrt{x}$ at negative numbers, and at 0, direct substitution because the function isn't continuous (or even defined) there.

Catalog Check 🎯

Substitute or Stop? 🔽

\lim_{x \to a} g(x) = M

Law	Statement
Sum	$\lim (f + g) = L + M$
Constant Multiple	$\lim (c \cdot f) = cL$
Product	$\lim (fg) = LM$
Quotient	$\lim \dfrac{f}{g} = \dfrac{L}{M}$ if
Power	$\lim f^n = L^n$ for positive integers $n$

💡 These laws let you take a complicated expression apart, evaluate each piece, and combine — exactly what direct substitution does in shorthand.

🔁 Composition of Continuous Functions

If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$ , then $\lim_{x \to a} f(g(x)) = f(g(a)).$

In words: "you can move the limit inside a continuous function."

Worked example: $\lim_{x \to 3} \sqrt{x^2 + 7}$ .

Inside: $\lim_{x \to 3} (x^2 + 7) = 16$ .
Outside: $\sqrt{\,\cdot\,}$ is continuous at 16.

📝 Examples Combining the Laws

Ex 1. $\lim_{x \to 2} \bigl[(x^3 - 1)(x + 5)\bigr] = (8 - 1)(7) = 49$ . (Product law.)

Ex 2. $\lim_{x \to 0} \dfrac{e^x + 4}{\cos x + 2} = \dfrac{1 + 4}{1 + 2} = \dfrac{5}{3}$ . (Quotient law.)

Ex 3. $\lim_{x \to \pi/2} \sin(\cos x) = \sin(\cos(\pi/2)) = \sin 0 = 0$ . (Composition.)

Apply the Laws 🎯

Combine the Laws 🧮

Given $\lim_{x \to a} f = 5$ and $\lim_{x \to a} g = 4$ :

1) $\lim_{x \to a} (f + g) = ?$

2) $\lim_{x \to a} (3fg) = ?$

3) $\lim_{x \to a} \dfrac{f^2}{g} = ?$ (decimal or fraction)

💡 "Indeterminate" doesn't mean "no answer." It means "more work needed."

🗺️ What to Do Next (A Roadmap)

Indeterminate form	Most common technique
$0/0$ , polynomial-only	Factor and cancel the common factor
$0/0$ , contains $\sqrt{}$	Rationalize with the conjugate
$0/0$ , contains $\sin$ or $\cos$	Use known special limits like $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$
$\infty / \infty$	Divide top and bottom by highest power of $x$
$\infty - \infty$	Combine into a single fraction

🔑 The next four micro-topics in this unit develop these techniques one at a time.

🧠 Substitution vs. Indeterminate vs. DNE

Three cleanly different outcomes when you "plug in":

Number — direct substitution wins, you're done.
Indeterminate form — you don't have the answer yet; switch to factoring/rationalizing/etc. The limit might still exist.
Nonzero over zero — the limit DNE because of unbounded behavior; sometimes we can describe it as $\pm\infty$ .

⚠️ Common mistake: writing "DNE" when you actually got $0/0$ . $0/0$ is not an answer — it's a request to do more work.

Diagnose the Trigger 🎯

Conclusion:

\sqrt{16} = 4

Clean number $L$	Limit is $L$ ✅	Done
Nonzero / 0	Vertical asymptote — limit DNE (or $\pm\infty$ )	Analyze sign on each side
$0/0$	Indeterminate	Factor, rationalize, L'Hôpital, or known limit
$\infty - \infty$	Indeterminate	Combine into a single fraction first
$0 \cdot \infty$	Indeterminate	Rewrite as a quotient, then apply techniques
$\infty / \infty$	Indeterminate	Compare growth rates / divide by dominant term