2 Limits and Continuity

Understanding the meaning of a limit

A limit describes nearby behavior rather than necessarily reporting the function's value at the target. The notation

means that $f(x)$ approaches $L$ as $x$ approaches $a$. The value $f(a)$ may equal $L$, differ from $L$, or be undefined.

For example, consider

Factoring gives

Therefore,

The function is undefined at $x=1$, but its nearby values approach $2$. This produces a Removable discontinuity, represented graphically by a hole at $(1,2)$.

Takeaway: Separate the question “What value does the function have at the point?” from the question “What value do nearby function values approach?”

Evaluating limits

Limits can be estimated with a table, interpreted from a graph, or found exactly with algebra.

A numerical table uses input values on both sides of the target. For $f(x)=x^2+1$, values near $x=2$ approach $5$, suggesting

Numerical evidence is useful for estimation, but rounded values or inputs that are not sufficiently close may hide important behavior.

From a graph, trace the function toward the target from the left and from the right. If both sides approach the same height, the One-sided limit values agree and the two-sided limit exists. A hole does not prevent a limit from existing.

For exact algebraic evaluation, use these strategies:

• Substitute directly when the function is continuous at the target. For example,
  \

  $$\lim_{x\to3}(2x^2-x+4)=2(3)^2-3+4=19.$$

• If substitution produces $\frac{0}{0}$, factor and cancel common factors before substituting. For example,
  \

  $$\lim_{x\to2}\frac{x^2-4}{x-2}=\lim_{x\to2}(x+2)=4.$$

• For radicals, multiply by the conjugate. For example,
  \

  $$\lim_{x\to0}\frac{\sqrt{x+1}-1}{x}=\lim_{x\to0}\frac{1}{\sqrt{x+1}+1}=\frac{1}{2}.$$

Limit laws permit sums, differences, constant multiples, products, quotients, and powers to be evaluated from the corresponding individual limits when those limits exist. For a quotient, the limiting denominator must be nonzero.

Takeaway: Use tables and graphs to build intuition, but use algebraic simplification when an exact value is required.

Comparing behavior from both sides

A One-sided limit records behavior from one direction:

uses values less than $a$, while

uses values greater than $a$. A two-sided limit exists exactly when both one-sided limits exist and are equal:

Consider

The one-sided limits are

Because they agree, the two-sided limit is $3$, even though the formula changes at $x=2$. By contrast, if the left-hand limit is $2$ and the right-hand limit is $5$, the two-sided limit does not exist.

Takeaway: Always compare the left and right sides before assigning a two-sided limit.

Infinite limits and vertical behavior

An infinite limit describes unbounded behavior; $\infty$ and $-\infty$ are not finite real-number values. For example,

Since the one-sided behaviors differ, the two-sided limit at $x=0$ does not exist. The line $x=0$ is a Vertical asymptote.

A denominator that becomes zero may signal a vertical asymptote, but cancellation must be checked first. The expression

has a hole at $x=1$, not a vertical asymptote. In contrast, $f(x)=\frac{1}{x-1}$ becomes unbounded near $x=1$, so $x=1$ is a vertical asymptote.

Takeaway: Distinguish a hole caused by a canceled factor from unbounded behavior caused by a remaining denominator factor.

Limits at infinity and end behavior

Limits as $x$ tends to $\infty$ or $-\infty$ describe end behavior:

For example,

Thus, $y=3$ is a Horizontal asymptote.

For a rational function $f(x)=\frac{p(x)}{q(x)}$, compare the degrees of the numerator and denominator:

• If $\deg p<\deg q$, the horizontal asymptote is $y=0$.

• If $\deg p=\deg q$, the horizontal asymptote is the ratio of the leading coefficients.

• If $\deg p>\deg q$, there is generally no horizontal asymptote; polynomial division may reveal a slant or higher-degree polynomial asymptote.

For example,

because dividing numerator and denominator by $x^2$ gives

An asymptote describes limiting behavior, so a graph may cross it.

Takeaway: Degree comparison is a fast way to predict the end behavior of a rational function.

Continuity and discontinuities

Continuity at $x=a$ requires three conditions:

1. $f(a)$ is defined.

2. $\lim_{x\to a}f(x)$ exists.

3. $\lim_{x\to a}f(x)=f(a)$.

Equivalently,

A Removable discontinuity occurs when the limit exists but the function is missing a value or has the wrong value. A jump discontinuity occurs when the one-sided limits exist but are unequal. An infinite discontinuity occurs when the function becomes unbounded near the point.

Polynomials are continuous everywhere. Rational functions are continuous wherever their denominators are nonzero, and exponential, logarithmic, trigonometric, and root functions are continuous on their domains.

For a piecewise function, test the boundary between formulas. If

then

All three continuity conditions hold at $x=1$.

Takeaway: To test continuity at a boundary, compare the left-hand limit, right-hand limit, and assigned function value.

Using continuity to guarantee values

The Intermediate Value Theorem connects continuity with guaranteed existence. If a function is continuous on the closed interval $[a,b]$, it takes every value between $f(a)$ and $f(b)$. In particular, if the endpoint values have opposite signs, there is at least one $c\in(a,b)$ such that $f(c)=0$.

For

continuity holds because this is a polynomial. The endpoint values are

Since zero lies between $-1$ and $5$, the theorem guarantees a number $c\in(1,2)$ for which

The theorem establishes that a solution exists even when an exact algebraic expression is unavailable. It does not by itself identify the solution or prove that the solution is unique.

Takeaway: Continuity prevents a function from skipping intermediate values, making sign changes a powerful existence test.