01 / 21

Defining Limits
and Limit Notation

Explore the foundational concept of calculus: how functions behave near a point, even when they can't reach it.

AP Calculus BC

What Is a Limit?

A limit describes the value that a function \( f(x) \) approaches as the input \( x \) gets closer and closer to some number \( c \). We are not asking what happens at \( c \): we are asking what happens near \( c \).

Limit Notation Explained

The standard notation for a limit is:

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

\( x \to c \)
\( x \) approaches the value \( c \) but does not equal \( c \)
\( f(x) \)
The function whose behavior we are analyzing near \( c \)
\( L \)
The value the outputs approach: the limit value

Visualizing a Limit on a Graph

x y from left from right c L f(c)

The Informal Definition

"Arbitrarily close" means: no matter how tiny a distance you name, the function values will eventually stay within that distance of \( L \) as \( x \) nears \( c \).

The \( \varepsilon \)-\( \delta \) Idea

x y L
\(L+\varepsilon\)
\(L-\varepsilon\)
 c
\(c-\delta\)
\(c+\delta\)

One-Sided Limits

We can approach \( c \) from only one direction. This gives us two one-sided limits:

Left-Hand Limit
\( \displaystyle\lim_{x \to c^-} f(x) \): approach \( c \) from values less than \( c \)
Right-Hand Limit
\( \displaystyle\lim_{x \to c^+} f(x) \): approach \( c \) from values greater than \( c \)

Limit vs. Function Value

\( \displaystyle\lim_{x \to c} f(x) \) \( f(c) \)
What it asks Where is \( f(x) \) heading? What is \( f \) at \( c \)?
Requires \( f(c) \) defined? No Yes
Uses values at \( x = c \)? No: only nearby Yes: exactly at \( c \)
Can they differ? Yes: the limit can equal \( 3 \) even if \( f(c) = 7 \)

When Limits Do Not Exist

Jump
Left and right limits exist but are not equal: \( \lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x) \)
Unbounded
\( f(x) \) increases or decreases without bound as \( x \to c \) (vertical asymptote)
Oscillation
\( f(x) \) oscillates infinitely often near \( c \), e.g. \( \sin(1/x) \) as \( x \to 0 \)

How to Evaluate a Limit

1
Direct substitution: Plug \( c \) into \( f(x) \). If you get a real number, that is the limit.
2
Indeterminate form? If you get \( \frac{0}{0} \), try algebraic simplification: factor, rationalize, or cancel.
3
One-sided check: Evaluate \( \lim_{x\to c^-} \) and \( \lim_{x\to c^+} \) separately. If they disagree, the limit DNE.
4
Numerical/graphical: Build a table of values or inspect the graph to confirm your answer.

Example 1: Direct Substitution

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

1
Substitute \( x = 2 \) directly: \( 3(2)^2 - 5(2) + 1 \).
2
Calculate: \( 3(4) - 10 + 1 = 12 - 10 + 1 = \) \( 3 \).
3
Since the function is a polynomial (continuous everywhere), direct substitution works.

Example 2: The \( \frac{0}{0} \) Form

$$ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} $$

1
Substitute \( x = 3 \): \( \frac{9-9}{3-3} = \frac{0}{0} \). Indeterminate!
2
Factor the numerator: \( x^2 - 9 = (x-3)(x+3) \).
3
Cancel \( (x-3) \): \( \displaystyle\frac{(x-3)(x+3)}{x-3} = x + 3 \) for \( x \neq 3 \).
4
Substitute into simplified form: \( 3 + 3 = \) \( 6 \).

Example 3: Reading a Graph

x y 1 2 3 1 2 Limit = 2 f(2) = 1

Confirming with a Table

Evaluate \( \displaystyle\lim_{x \to 0} \frac{\sin x}{x} \) by checking values near \( x = 0 \):

\( x \) \( -0.1 \) \( -0.01 \) \( -0.001 \) 0 \( 0.001 \) \( 0.01 \) \( 0.1 \)
\( \frac{\sin x}{x} \) 0.9983 0.99998 0.9999998 undef 0.9999998 0.99998 0.9983

Common Mistakes

✕ Plugging in blindly
Substituting \( x = c \) when the function is undefined there and calling \( \frac{0}{0} \) the answer instead of simplifying.
✕ Confusing limit with value
Assuming \( \lim_{x \to c} f(x) = f(c) \). This is only true when \( f \) is continuous at \( c \).
✕ Ignoring one side
Checking only the left or right approach and declaring the limit exists without verifying both sides agree.

Summary of Key Ideas

Concept Notation Key Point
Two-sided limit \( \lim_{x \to c} f(x) = L \) Both sides must agree
Left-hand limit \( \lim_{x \to c^-} f(x) \) Approach from below
Right-hand limit \( \lim_{x \to c^+} f(x) \) Approach from above
DNE (jump) One-sided limits differ Two-sided limit fails
\( \varepsilon \)-\( \delta \) \( 0 < |x-c| < \delta \Rightarrow |f(x)-L| < \varepsilon \) Formal precision

Question 1 of 5

What does \( \displaystyle\lim_{x \to 4} f(x) = 7 \) tell us?

Question 2 of 5

Evaluate: \( \displaystyle\lim_{x \to -1} \frac{x^2 - 1}{x + 1} \)

Question 3 of 5

If \( \displaystyle\lim_{x \to 2^-} f(x) = 5 \) and \( \displaystyle\lim_{x \to 2^+} f(x) = 3 \), what is \( \displaystyle\lim_{x \to 2} f(x) \)?

Question 4 of 5

Which statement is always true if \( \displaystyle\lim_{x \to c} f(x) = L \)?

Question 5 of 5

\( \displaystyle\lim_{x \to 0} \frac{\sin x}{x} = \) ?