2. Limits: A Study Guide

Understanding What a Limit Describes

A limit describes the output value a function approaches as its input gets close to a specified number. The notation $\lim_{x\to a}f(x)=L$ means that $f(x)$ approaches $L$ as $x$ approaches $a$. The input need not equal $a$, and the function need not be defined at $a$.

A removable discontinuity

Consider

At $x=1$, the original expression is undefined. For $x\ne1$, however,

Therefore,

The graph has a hole at $(1,2)$, but the limit exists because nearby values approach $2$. This distinction between a function's value and its limit is fundamental: the limit concerns nearby behavior, not necessarily the value at the target input.

Estimating from tables and graphs

Values close to the target can suggest a limit. For the function above, values at $x=0.9,0.99,1.01,1.1$ are respectively $1.9,1.99,2.01,2.1$, which indicates approach toward $2$. On a graph, follow the curve toward the target from both sides. If both sides approach the same height, that height is the two-sided limit; a missing point does not prevent the limit from existing.

Takeaway: A limit describes nearby behavior and can exist even when the function is undefined at the point.

Comparing Left- and Right-Hand Behavior

To determine a two-sided limit, compare behavior from the left and from the right. The one-sided limit $\lim_{x\to a^-}f(x)$ examines inputs less than $a$, while $\lim_{x\to a^+}f(x)$ examines inputs greater than $a$.

The existence test

The relationship is

Both one-sided limits must exist and agree. If they approach different values, the two-sided limit does not exist.

Example: a jump

Let

Then

Because the one-sided limits disagree, $\lim_{x\to2}f(x)$ does not exist. This approach is especially useful for piecewise functions and for analyzing behavior near a possible vertical asymptote.

A limit can also fail because the function grows without bound or oscillates without settling toward one value. In every case, the value assigned exactly at $x=a$ does not determine the two-sided limit.

Takeaway: Always check both sides when a function is piecewise, discontinuous, or undefined near the target.

Making Limits Precise with Epsilon and Delta

The intuitive idea that outputs get arbitrarily close to $L$ can be expressed precisely by the epsilon-delta definition. The statement

means that for every $\varepsilon>0$, there is a $\delta>0$ such that

The quantity $\varepsilon$ is the permitted output error. The quantity $\delta$ is the input distance that guarantees this error bound. The condition $0<|x-a|$ excludes the point $x=a$, which is why the value $f(a)$ is irrelevant to the limit.

Example: a linear function

To prove

rewrite the output difference:

Given any $\varepsilon>0$, choose

If $0<|x-3|<\delta$, then

This verifies the limit. For a right-hand limit, the input condition is $0<x-a<\delta$; for a left-hand limit, it is $-\delta<x-a<0$.

Takeaway: The formal definition turns “gets close” into a quantified guarantee: every output tolerance can be achieved by choosing a suitable input tolerance.

Using Limit Laws and Algebraic Simplification

When the component limits exist, limit laws allow a complicated expression to be broken into simpler parts. If

then sums, differences, constant multiples, products, powers, and appropriate roots follow the corresponding algebraic rules. For quotients,

requires $M\ne0$.

Direct substitution

Polynomials can be evaluated by direct substitution:

For a rational function whose denominator does not approach zero,

Direct substitution is not a complete method when it produces an indeterminate form such as $0/0$. That result signals the need for additional algebraic or analytic work.

Factoring and canceling

For

factor the numerator:

Thus,

The cancellation preserves the behavior near $x=3$, even though the original expression remains undefined at that point.

Rationalizing with a conjugate

For

multiply by the conjugate:

for $x\ne0$. Therefore, the limit equals $1/2$.

Takeaway: Use direct substitution when valid; when it gives $0/0$, simplify without changing the nearby behavior.

Applying the Squeeze Theorem

The Squeeze Theorem is useful when a function is difficult to evaluate directly but can be bounded by simpler functions. If

near $a$, and

then

Oscillation controlled by shrinking bounds

Because

multiplication by $|x|$ gives

As $x\to0$, both $-|x|$ and $|x|$ approach $0$. The Squeeze Theorem therefore yields

The middle function oscillates increasingly rapidly, but its amplitude shrinks toward zero, forcing its values toward zero.

Takeaway: Matching bounds can establish a limit even when the function itself is oscillatory or awkward to simplify.

Analyzing Infinite Behavior and Asymptotes

An infinite limit describes unbounded behavior as the input approaches a finite number. For example,

means that the function eventually exceeds every prescribed positive bound as $x$ approaches zero from the right. It does not mean that the function reaches a real value called infinity. From the left,

Since the two sides have different behavior, there is no two-sided infinite limit at zero for this function.

A vertical asymptote is a line $x=a$ where at least one one-sided limit is infinite. For

both one-sided limits as $x\to-3$ are $+\infty$, so $x=-3$ is a vertical asymptote. To determine the sign near such a line, inspect the signs of the numerator and denominator on each side of the critical input.

Limits at infinity

A limit at infinity describes end behavior:

means that the function approaches $L$ as $x$ increases without bound. The analogous notation with $x\to-\infty$ describes behavior as $x$ decreases without bound. If either limit equals $L$, then $y=L$ is a horizontal asymptote.

For a rational function $f(x)=p(x)/q(x)$:

• If the degree of $p$ is less than the degree of $q$, the horizontal asymptote is $y=0$.

• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

• If the degree of $p$ is greater, this comparison does not produce a horizontal asymptote; polynomial or oblique behavior may occur.

For example,

so $y=3$ is a horizontal asymptote. Also,

so $y=0$ is a horizontal asymptote. A graph may cross a horizontal asymptote; the line describes end behavior rather than a forbidden intersection.

Takeaway: Infinite limits describe unbounded local behavior, while limits at infinity describe long-run behavior.

A Practical Workflow for Finding Limits

A reliable procedure helps organize limit problems.

Decision process

1. Try direct substitution.

2. If the result is a defined finite expression, use it.

3. If the result is $0/0$, factor, cancel, rationalize, or combine fractions.

4. If the function is piecewise or discontinuous, calculate the left-hand and right-hand limits separately.

5. If the denominator approaches zero while the numerator does not, analyze signs on both sides to determine possible infinite limits.

6. If the input approaches $\infty$ or $-\infty$, compare dominant terms, especially the highest powers in a rational function.

7. Interpret the result graphically: a hole indicates a removable discontinuity, unequal one-sided limits indicate a jump, and infinite behavior indicates a vertical asymptote.

Connections across methods

Tables and graphs are useful for estimating behavior and forming a conjecture. Algebraic simplification can establish exact values for many expressions. One-sided analysis handles directional behavior, while the epsilon-delta definition supplies the foundational standard for a rigorous finite limit. The same concept therefore connects numerical, graphical, algebraic, and formal reasoning.

Final checklist: Ask what the input approaches, test direct substitution, identify any indeterminate form, choose an appropriate simplification or comparison, check both sides when necessary, and interpret the result in terms of the graph.