What is the difference between a limit not existing and being infinite

In calculus a limit describes the value a function approaches as the input gets arbitrarily close to a point. If the function settles near a single number, the limit exists and equals that number. When the behavior is different, the limit either does not exist (DNE) or is said to be infinite. Recognizing the distinction is essential for correctly applying limit laws on exams.

When a limit does not exist

A limit does not exist when the function fails to approach any single value, finite or infinite. This can happen because the left‑hand and right‑hand limits disagree, or because the function oscillates without settling. For instance, $\lim_{x\to0}\sin\frac{1}{x}$ has no limit because the values swing between –1 and 1 as $x$ approaches zero from either side. A classic DNE case is when the left‑hand limit equals 1 while the right‑hand limit equals –1, such as $\lim_{x\to0}\frac{|x|}{x}$.

Typical ways a limit can fail to exist:

• Left‑hand and right‑hand limits are different (e.g., $\lim_{x\to0}\frac{|x|}{x}$)
• Function oscillates without settling (e.g., $\lim_{x\to0}\sin\frac{1}{x}$)
• Approaches different finite values along two sequences (e.g., rational vs irrational approaches)
• Unbounded behavior that is not consistently positive or negative (e.g., $\lim_{x\to0}\frac{\sin x}{x}$ stays bounded but does not settle)

Infinite limits

An infinite limit means the function grows without bound in a specific direction; the values become arbitrarily large (or small) as the input approaches the point. The limit is written as $\lim_{x\to a}f(x)=\infty$ or $-\infty$. A classic example is $\lim_{x\to0^{+}}\frac{1}{x}=+\infty$, where the function’s values exceed any prescribed number as $x$ gets positive and close to zero. Infinite limits often appear in asymptote analysis, indicating a vertical asymptote at the point where the function blows up.

Procedure to decide whether a limit is infinite:

1. 1Compute the one‑sided expressions if needed.
2. 2Show that for every large $M>0$ there exists $\delta>0$ such that $0<|x-a|<\delta$ implies $f(x)>M$ (or $f(x)<-M$).
3. 3If such a $\delta$ can be found, the limit is $+\infty$ (or $-\infty$).
4. 4If you cannot bound the function in this way, the limit is not infinite.

Comparison of DNE and infinite limits: