Direct Substitution Method

🎯 Direct Substitution — The Idea

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

Topics in This Part

🔑 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:

1. Polynomials are continuous everywhere.
2. Rational functions are continuous everywhere their denominator is nonzero.
3. $\sin x$, $\cos x$, $e^x$ are continuous everywhere.
4. $\ln x$ is continuous on $x > 0$.
5. $\sqrt{x}$ is continuous on $x \ge 0$.
6. 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.

📝 Easy Examples

Concept Check 🎯

Substitute 🧮

Compute each limit by direct substitution:

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

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

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

✅ When Direct Substitution Works

Part 2 of 4 — A continuity catalog

Topics in This Part

🔑 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 everywhere.

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 $q(x) \ne 0$.

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

🌀 Trig, Exp, Log, Roots

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

Catalog Check 🎯

Substitute or Stop? 🔽

📜 Limit Laws — The Algebra of Substitution

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

Topics in This Part

🔑 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 $\lim_{x \to a} g(x) = M$ (both exist), and $c$ is a constant:

💡 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.
• Conclusion: $\sqrt{16} = 4$.

📝 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)

🚦 When Direct Substitution Fails

Part 4 of 4 — Recognize the trigger and switch tools

Topics in This Part

🔑 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:

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

🗺️ What to Do Next (A Roadmap)

🔑 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":

1. Number — direct substitution wins, you're done.
2. Indeterminate form — you don't have the answer yet; switch to factoring/rationalizing/etc. The limit might still exist.
3. 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 🎯

Exit Check ✅