Indeterminate vs. Undefined Limits: 0/0, Infinity and DNE

Undefined vs. indeterminate vs. DNE

Undefined function value

The expression cannot be evaluated at the specified input. For example, \(1/x\) is undefined at \(x=0\). This says nothing by itself about nearby behavior.

Indeterminate form

The separate limiting behaviors do not determine the result. For example, \(0/0\) requires further analysis; it is not an answer and does not automatically mean DNE.

Limit does not exist (DNE)

The function fails to approach a single finite number. Different one-sided limits and persistent oscillation are common reasons.

Infinite limit

The values grow without bound in a specified direction. Writing \(+\infty\) or \(-\infty\) describes that behavior; infinity is not a finite real-number limit. Some courses classify these as DNE when asking specifically for a finite limit, so state the behavior explicitly.

A limit concerns values near the input, not the value at the input. If you need a refresher on the notation, start with Introduction to Limits.

Which forms are indeterminate?

The standard forms are \(0/0\), \(\infty/\infty\), \(0\cdot\infty\), \(\infty-\infty\), \(0^0\), \(1^\infty\), and \(\infty^0\). They describe limits of expressions, rather than arithmetic operations on infinity. In particular, \(0^0\) is an indeterminate limit form; it should not appear on a separate list of forms proving DNE.

For instance, as \(x\to0\), both \(x/x\) and \(x^2/x\) have the form \(0/0\), yet their limits are 1 and 0. The form alone cannot choose between these answers. Even DNE is possible: \(|x|/x\) has the same form, but approaches −1 from the left and 1 from the right.

Try factoring, rationalizing, or rewriting first. L’Hôpital’s rule applies to suitable quotients of type \(0/0\) or \(\infty/\infty\) only when its differentiability and derivative conditions hold. A product, difference, or power needs an appropriate transformation before using the rule; do not differentiate the top and bottom of every fraction automatically.

Worked examples: undefined at the point, but a limit exists

A removable hole gives a finite limit

Evaluate \(\lim_{x\to2}\frac{x^2-4}{x-2}\). Substitution gives \(0/0\), so factor:

[begin{aligned}frac{x^2-4}{x-2}&=frac{(x-2)(x+2)}{x-2}\&=x+2quad(xne2).end{aligned}]

Therefore the limit is \(2+2=4\). The original formula remains undefined at 2; canceling the factor does not assign a value to that hole.

A zero times an unbounded quantity can approach zero

For \(x\to0^+\), the expression \(x\ln x\) has the indeterminate form \(0\cdot(-\infty)\). Rewrite it as \(\ln x/(1/x)\). The quotient has type \((-\infty)/(+\infty)\), and L’Hôpital’s rule gives:

\[\lim_{x\to0^+}\frac{\ln x}{1/x}=\lim_{x\to0^+}\frac{1/x}{-1/x^2}=\lim_{x\to0^+}(-x)=0.\]

The derivatives exist for \(x>0\), the denominator derivative is nonzero, and the derivative quotient has a limit. Those checks justify the rule here.

When does a limit fail to exist?

Different behavior from the left and right

For \(1/x\) at 0, \(\lim_{x\to0^-}1/x=-\infty\), while \(\lim_{x\to0^+}1/x=+\infty\). The two-sided limit is DNE. By contrast, \(1/x^2\) tends to \(+\infty\) from both sides: it has a positive infinite limit, though no finite real limit. Simply writing “a nonzero number divided by zero” misses this distinction.

Oscillation that does not settle

The limit \(\lim_{x\to0}\sin(1/x)\) is DNE. Even restricting to \(x>0\), take \(x_n=1/(\pi/2+2\pi n)\) and \(y_n=1/(3\pi/2+2\pi n)\). Both sequences approach 0, but the function equals 1 along the first and −1 along the second. It cannot approach a single number.

Need more practice? Calculus for Beginners covers Calculus step by step, with practice problems and answers. Look inside · $17.99

Different paths in a multivariable limit

For \((x^2-y^2)/(x^2+y^2)\) as \((x,y)\to(0,0)\), the path \(y=0\) gives 1 and the path \(x=0\) gives −1. These conflicting results prove DNE. Agreement along just two paths would not prove that a multivariable limit exists.

Check your understanding

1. Find \(\lim_{x\to3}(x^2-9)/(x-3)\).
2. Describe \(\lim_{x\to0}5/x^2\), including whether it is finite.
3. Does \(\lim_{x\to0}|x|/x\) exist?

For your next problem, check the domain, try substitution, simplify an indeterminate form, and compare one-sided behavior when necessary. Use the free Calculus lessons, practice, and formula review to choose the next skill to study.

Reference: OpenStax, The Limit of a Function and L’Hôpital’s Rule, for definitions, one-sided behavior, and the rule’s conditions.

What does DNE mean in math?

DNE stands for does not exist. You will see it most often as an answer to a limit problem. It means the function does not approach one single finite number as x gets close to the target value. There are three common reasons:

• The left and right sides approach different values. For |x| ÷ x at x = 0, the left side gives −1 and the right side gives 1.
• The function keeps oscillating. sin(1/x) swings between −1 and 1 forever near x = 0.
• The function grows without bound in different directions. 1/x heads to −∞ from the left and +∞ from the right.

DNE also appears outside limits. A derivative DNE at a sharp corner, like |x| at x = 0. The slope of a vertical line DNE. Teachers handle infinite limits differently. Some write ∞ and others write DNE, so follow your class’s convention.

Undefined vs. indeterminate vs. DNE at a glance

Frequently Asked Questions

Is indeterminate the same as undefined?

No. Undefined means an expression has no value, like 7 ÷ 0. Indeterminate describes a limit form, such as 0/0 or ∞/∞, that does not tell you the answer yet. For example, (x² − 9)/(x − 3) gives 0/0 at x = 3, but after factoring and canceling it becomes x + 3, so the limit is 6.

What does DNE mean in math?

DNE stands for does not exist. In calculus it means a limit does not approach a single finite number. Common reasons are a jump, where the left and right limits differ, or endless oscillation, like sin(1/x) near 0. Some teachers also write DNE for limits that grow to infinity, so check how your class wants those labeled.

Is DNE the same as undefined?

Not quite. Undefined describes a function value: f(a) has no output. DNE describes a limit: the function does not approach one number near a. They can happen separately. The function x²/x is undefined at x = 0, yet its limit there is 0, so the limit exists even though the value does not.

Can a limit be undefined?

Strictly speaking, a limit either exists or does not exist, so DNE is the better word. For 1/x as x approaches 0, the left side heads to −∞ and the right side to +∞, so the limit does not exist. For 1/x², both sides go to +∞, and many courses write the limit as ∞ to describe that unbounded behavior.

Is 0/0 DNE?

No, 0/0 is not automatically DNE. It is an indeterminate form, a signal to do more work by factoring, simplifying, rationalizing or using L’Hôpital’s rule. The limit of sin x / x as x approaches 0 is 1, even though substitution gives 0/0. The final limit could be a number, infinite, or nonexistent.

Can an indeterminate limit not exist?

Yes. An indeterminate form only says the answer is not decided yet, and sometimes the final answer is DNE. For x/|x| at x = 0, substitution gives 0/0, but the right-hand limit is 1 and the left-hand limit is −1. Since the one-sided limits disagree, the two-sided limit does not exist.