Introduction to Limits

🎯 What Is a Limit?

Part 1 of 7

The Big Idea

A limit describes what a function approaches as $x$ gets closer to a value — even if it never gets there!

$\lim_{x \to c} f(x) = L$

"As $x$ approaches $c$, $f(x)$ approaches $L$."

Why Limits Matter

Limits are the foundation of calculus:

• Derivatives = limits of difference quotients
• Integrals = limits of Riemann sums
• Continuity = defined using limits

Intuitive Example

$f(x) = \frac{x^2-1}{x-1}$

At $x = 1$: $f(1) = 0/0$ — undefined!

But simplify: $f(x) = \frac{(x-1)(x+1)}{x-1} = x+1$ (when $x \neq 1$).

As $x \to 1$, $f(x) \to 2$. So $\lim_{x \to 1} f(x) = 2$.

The limit exists even though $f(1)$ is undefined!

📊 Estimating Limits from Tables

Approach from Both Sides

For $f(x) = \frac{x^2-1}{x-1}$:

Both sides approach 2. ✓

When Sides Disagree

$g(x) = \begin{cases} x+1 & x < 2 \\ x+3 & x \geq 2 \end{cases}$

From left: $g(x) \to 3$. From right: $g(x) \to 5$.

Since $3 \neq 5$, the limit does not exist at $x = 2$.

💡 Both one-sided limits must agree for the (two-sided) limit to exist.

📝 One-Sided Limits

Left-Hand Limit (from below)

$\lim_{x \to c^-} f(x) = L$

$x$ approaches $c$ from values less than $c$.

Right-Hand Limit (from above)

$\lim_{x \to c^+} f(x) = L$

$x$ approaches $c$ from values greater than $c$.

The Connection

$\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$

Example

$h(x) = |x|/x$

• $\lim_{x \to 0^-} = -1$ (negative values)
• $\lim_{x \to 0^+} = 1$ (positive values)
• $\lim_{x \to 0}$ DNE (they disagree)

Limits Intuition Quiz 🎯

Evaluate Limits 🧮

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

2) $\lim_{x \to 4} \sqrt{x}$ = ?

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

Limit Concepts 🔽

Exit Quiz ✅

🧮 Limit Laws

Part 2 of 7

Basic Limit Laws

If $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$:

Direct Substitution

For polynomials and rational functions (where defined):

$\lim_{x \to c} p(x) = p(c)$

Just plug in! This works for any continuous function.

📝 Using the Laws

Example 1: Sum and Power

$\lim_{x \to 2} (x^3 + 4x) = 2^3 + 4(2) = 8 + 8 = 16$

Example 2: Quotient

$\lim_{x \to 3} \frac{x^2+1}{x-1} = \frac{9+1}{3-1} = \frac{10}{2} = 5$

Example 3: Root Law

$\lim_{x \to 9} \sqrt{x+7} = \sqrt{9+7} = \sqrt{16} = 4$

Example 4: Combined Laws

$\lim_{x \to 1} \frac{(2x+1)^3}{\sqrt{x+3}} = \frac{(3)^3}{\sqrt{4}} = \frac{27}{2}$

💡 Direct substitution is the first thing to try. Only use algebraic techniques when substitution gives $0/0$.

🤏 The Squeeze Theorem

Statement

If $g(x) \leq f(x) \leq h(x)$ near $x = c$, and

$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$

then $\lim_{x \to c} f(x) = L$.

Classic Example

$\lim_{x \to 0} x^2 \sin(1/x)$

We know $-1 \leq \sin(1/x) \leq 1$, so:

$-x^2 \leq x^2 \sin(1/x) \leq x^2$

Since $\lim_{x \to 0} (-x^2) = 0$ and $\lim_{x \to 0} x^2 = 0$:

$\lim_{x \to 0} x^2 \sin(1/x) = 0$

Squeezed between two functions that both go to 0!

Limit Laws Quiz 🎯

Apply Limit Laws 🧮

1) $\lim_{x \to -1} (2x^3 + 5)$ = ?

2) $\lim_{x \to 5} \frac{x+1}{x-3}$ = ?

3) $\lim_{x \to 0} \sqrt{4-x}$ = ?

Limit Properties 🔽

Exit Quiz ✅

🔧 Algebraic Limit Techniques

Part 3 of 7

When Direct Substitution Gives $0/0$

$0/0$ is indeterminate — the limit could be any value. We need algebraic manipulation.

Technique 1: Factor and Cancel

$\lim_{x \to 3} \frac{x^2-9}{x-3} = \lim \frac{(x-3)(x+3)}{x-3} = \lim(x+3) = 6$

Technique 2: Rationalize (Multiply by Conjugate)

$\lim_{x \to 0} \frac{\sqrt{x+4}-2}{x}$

Multiply by $\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}$:

$= \lim \frac{(x+4)-4}{x(\sqrt{x+4}+2)} = \lim \frac{x}{x(\sqrt{x+4}+2)} = \lim \frac{1}{\sqrt{x+4}+2} = \frac{1}{4}$

📝 More Techniques

Technique 3: Common Denominator

$\lim_{x \to 0} \frac{\frac{1}{x+2}-\frac{1}{2}}{x}$

Combine fractions in numerator:

$= \lim \frac{\frac{2-(x+2)}{2(x+2)}}{x} = \lim \frac{-x}{2x(x+2)} = \lim \frac{-1}{2(x+2)} = -\frac{1}{4}$

Technique 4: Factor Higher-Degree Polynomials

$\lim_{x \to 2} \frac{x^3-8}{x-2} = \lim \frac{(x-2)(x^2+2x+4)}{x-2} = 4+4+4 = 12$

Recall: $a^3-b^3 = (a-b)(a^2+ab+b^2)$

Decision Tree

1. Try direct substitution
2. If $0/0$: factor, rationalize, or simplify
3. If $k/0$ ($k \neq 0$): limit is $\pm\infty$ or DNE

🌊 Special Trig Limits

The Two Famous Limits

$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1-\cos x}{x} = 0$

Using Them

$\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim \frac{\sin(3x)}{3x} \cdot 3 = 1 \cdot 3 = 3$

$\lim_{x \to 0} \frac{\tan x}{x} = \lim \frac{\sin x}{x} \cdot \frac{1}{\cos x} = 1 \cdot 1 = 1$

$\lim_{x \to 0} \frac{\sin(5x)}{\sin(3x)} = \lim \frac{\sin(5x)}{5x} \cdot \frac{3x}{\sin(3x)} \cdot \frac{5}{3} = 1 \cdot 1 \cdot \frac{5}{3} = \frac{5}{3}$

💡 The key: make the argument of sin match the denominator.

Algebraic Techniques Quiz 🎯

Evaluate 🧮

1) $\lim_{x \to 5} \frac{x^2-25}{x-5}$ = ?

2) $\lim_{x \to 0} \frac{\sin(4x)}{2x}$ = ?

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

Technique Selection 🔽

Exit Quiz ✅

♾️ Limits at Infinity

Part 4 of 7

What Happens as $x \to \infty$?

$\lim_{x \to \infty} f(x) = L$

The function approaches a horizontal asymptote at $y = L$.

Rational Functions: Degree Comparison

$\lim_{x \to \infty} \frac{a_nx^n + \ldots}{b_mx^m + \ldots}$

Examples

$\lim_{x \to \infty} \frac{3x^2+1}{5x^2-2} = \frac{3}{5}$

$\lim_{x \to \infty} \frac{x+1}{x^3} = 0$

$\lim_{x \to \infty} \frac{x^3}{2x+1} = \infty$

🔧 The Divide-by-Highest-Power Technique

Standard Method

$\lim_{x \to \infty} \frac{3x^2-x+4}{2x^2+5x-1}$

Divide every term by $x^2$:

$= \lim_{x \to \infty} \frac{3-\frac{1}{x}+\frac{4}{x^2}}{2+\frac{5}{x}-\frac{1}{x^2}} = \frac{3-0+0}{2+0-0} = \frac{3}{2}$

Key Fact Used

$\lim_{x \to \infty} \frac{k}{x^n} = 0 \quad (n > 0)$

Non-Rational Functions

$\lim_{x \to \infty} \frac{\sqrt{4x^2+1}}{x} = \lim \frac{\sqrt{4x^2+1}}{x} = \lim \sqrt{\frac{4x^2+1}{x^2}} = \sqrt{4} = 2$

💡 For $x \to -\infty$, be careful: $\sqrt{x^2} = |x| = -x$ when $x < 0$!

📊 Infinite Limits (Vertical Asymptotes)

$\lim_{x \to c} f(x) = \pm\infty$

This is not a real limit (∞ is not a number), but we use the notation.

Finding Vertical Asymptotes

$f(x) = \frac{1}{(x-2)^2}$

As $x \to 2$: denominator → $0^+$, numerator → $1$.

$\lim_{x \to 2} \frac{1}{(x-2)^2} = +\infty$

Sign Analysis

$g(x) = \frac{1}{x-3}$

• $\lim_{x \to 3^+} = +\infty$ (denominator is small and positive)
• $\lim_{x \to 3^-} = -\infty$ (denominator is small and negative)
• $\lim_{x \to 3}$ DNE (one-sided limits disagree in sign)

Connection

• Horizontal asymptote: limit at infinity = finite
• Vertical asymptote: limit at finite point = infinity

Limits at Infinity Quiz 🎯

Evaluate 🧮

1) $\lim_{x \to \infty} \frac{7x^3}{x^3+x}$: leading coefficient ratio = ?

2) $\lim_{x \to \infty} \frac{2x+5}{x^2}$ = ? (enter 0 for zero)

3) The horizontal asymptote of $y = \frac{4x-1}{2x+3}$ is $y$ = ?

Asymptote Concepts 🔽

Exit Quiz ✅

📈 Limits from Graphs

Part 5 of 7

Reading Limits Graphically

Look at where the function HEADS, not where it IS.

Key Scenarios

Example: Function with a Hole

If the graph approaches $y=3$ from both sides at $x=2$, but there's an open circle at $(2,3)$ and a dot at $(2,5)$:

$\lim_{x \to 2} f(x) = 3 \quad \text{but} \quad f(2) = 5$

The limit and function value disagree — this is a removable discontinuity.

📊 Graph Reading Practice

Piecewise Example

Imagine a graph where:

• For $x < 1$: the curve approaches $y = 4$
• For $x > 1$: the curve approaches $y = 4$
• At $x = 1$: there's a filled dot at $(1, 2)$

Then:

• $\lim_{x \to 1^-} f(x) = 4$
• $\lim_{x \to 1^+} f(x) = 4$
• $\lim_{x \to 1} f(x) = 4$ ✓ (both sides agree)
• $f(1) = 2$ (actual value)

Vertical Asymptote Example

Near $x = 3$:

• From left: graph shoots up to $+\infty$
• From right: graph shoots down to $-\infty$

Then: $\lim_{x \to 3^-} = +\infty$, $\lim_{x \to 3^+} = -\infty$, $\lim_{x \to 3}$ DNE.

🔍 End Behavior from Graphs

Horizontal Asymptotes

If the graph levels off at $y = 2$ as $x \to \infty$:

$\lim_{x \to \infty} f(x) = 2$

Different Behavior at $\pm\infty$

Some functions (like $\arctan x$) have two horizontal asymptotes:

• $\lim_{x \to \infty} \arctan x = \pi/2$
• $\lim_{x \to -\infty} \arctan x = -\pi/2$

Common Mistakes

❌ Confusing the limit (where the graph heads) with the function value (where the dot is)

❌ Saying the limit is $\infty$ when it should be DNE (e.g., when one-sided limits are $+\infty$ and $-\infty$)

❌ Ignoring the difference between open circles (excluded) and closed circles (included)

Graph Limits Quiz 🎯

From the description: Graph has $f(2) = 7$, approaches $y=3$ from both sides as $x \to 2$.

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

2) $\lim_{x \to 2} f(x)$ = ?

3) $f(2)$ = ?

Graph Reading 🔽

Exit Quiz ✅

🎯 The Formal Definition of a Limit

Part 6 of 7

Epsilon-Delta Intuition

The formal statement of $\lim_{x \to c} f(x) = L$:

For every $\varepsilon > 0$, there exists $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \varepsilon$.

What This Means in Plain English

Translation: "Make $f(x)$ as close to $L$ as you want — I can make it happen by choosing $x$ close enough to $c$."

Worked Examples

Example 1: Prove $\lim_{x \to 3}(2x+1) = 7$

Need: $|f(x) - L| < \varepsilon$ when $0 < |x - c| < \delta$.

$|(2x+1) - 7| = |2x - 6| = 2|x - 3|$

We need $2|x-3| < \varepsilon$, so $|x-3| < \varepsilon/2$.

Choose $\delta = \varepsilon/2$. Then if $|x-3| < \delta$: $|(2x+1) - 7| = 2|x-3| < 2\delta = 2 \cdot \frac{\varepsilon}{2} = \varepsilon \; ✓$

Example 2: Prove $\lim_{x \to 1}(x^2) = 1$

$|x^2 - 1| = |x-1||x+1|$

Near $x=1$, if $|x-1| < 1$, then $0 < x < 2$, so $|x+1| < 3$.

Thus $|x^2-1| < 3|x-1|$. Choose $\delta = \min(1, \varepsilon/3)$.

The Intermediate Value Theorem — Preview

Statement

If $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a,b)$ with $f(c) = N$.

What It Says Intuitively

A continuous function can't skip values — it must hit every $y$-value between its endpoints.

Classic Application: Proving a Root Exists

To show $x^3 - x - 1 = 0$ has a solution in $[1,2]$:

• $f(1) = 1 - 1 - 1 = -1 < 0$
• $f(2) = 8 - 2 - 1 = 5 > 0$

Since $f$ is continuous and changes sign, IVT guarantees some $c \in (1,2)$ with $f(c) = 0$.

Why This Matters for Calculus

IVT is used to prove the Extreme Value Theorem and the Mean Value Theorem: the pillars of differential calculus.

Formal Limits Quiz 🎯

Find the delta:

1) For $\lim_{x \to 5}(3x) = 15$, if $\varepsilon = 0.06$, $\delta$ = ?

2) For $\lim_{x \to 2}(4x-1) = 7$, if $\varepsilon = 0.04$, $\delta$ = ?

3) $f(1)=-2, f(3)=4$, $f$ continuous. IVT guarantees a root in $(a,b)$. What is $a+b$?

Formal Definitions 🔽

Exit Quiz ✅

🏆 Limits — Complete Synthesis

Part 7 of 7

The Limit Evaluation Decision Tree

Given: Find lim f(x) as x→c
│
├─ Direct Substitution → works? → DONE
│
├─ 0/0 form? (Indeterminate)
│   ├─ Factor & cancel
│   ├─ Rationalize (conjugate)
│   ├─ Special trig limits
│   └─ L'Hôpital's Rule (calculus)
│
├─ k/0 form (k≠0)?
│   └─ Check signs → ±∞ or DNE
│
├─ x → ±∞?
│   ├─ Compare degrees
│   ├─ Divide by highest power of x
│   └─ Rationalize if radicals
│
└─ Piecewise / graph?
    └─ Check one-sided limits

Master Technique Reference

Algebraic Techniques Summary

One-sided vs Two-sided

$\lim_{x \to c} f(x) = L$ exists iff $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$.

Key Special Limits

$\lim_{x \to 0}\frac{\sin x}{x} = 1, \quad \lim_{x \to 0}\frac{1-\cos x}{x} = 0, \quad \lim_{x \to 0}\frac{e^x - 1}{x} = 1$

Bridge to Calculus

Limits → Derivatives

The derivative is DEFINED as a limit:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Every technique you've learned for evaluating limits directly applies to computing derivatives from the definition. The $0/0$ form is the essential challenge.

Limits → Integrals

The definite integral is DEFINED as a limit of Riemann sums:

$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\Delta x$

Limits at infinity power integral computation.

IVT → MVT → FTC

The Intermediate Value Theorem (IVT) leads to the Mean Value Theorem (MVT), which leads to the Fundamental Theorem of Calculus (FTC) — the crown jewel linking derivatives and integrals.

Master Limits Quiz 🎯

Mixed Limit Evaluation:

1) $\lim_{x \to 3}\frac{x^2 - 9}{x - 3}$ = ?

2) $\lim_{x \to \infty}\frac{4x + 1}{2x - 3}$ = ?

3) $\lim_{x \to 0}\frac{\sin(5x)}{x}$ = ?

Synthesis 🔽

Exit Quiz ✅