Infinite Limits: Vertical Asymptotes

Learning Objectives

By the end of this section you will be able to:

1. Describe what it means for a limit to be infinite — and why that is not the same as "the limit exists".
2. State the formal definition of $\displaystyle\lim_{x\to a}f(x)=+\infty$ and its one-sided cousins.
3. Identify vertical asymptotes of common functions by hand.
4. Classify the four canonical behaviors at a vertical asymptote.
5. Compute vertical asymptotes for rational functions algebraically.
6. Connect vertical asymptotes to exploding gradients in machine learning.

The Big Picture

In the previous section we watched $f(x)$ approach a finite height as $x\to\infty$. We now flip the script. The input stays finite — it slides toward some ordinary number $a$ — but the output refuses to settle anywhere. It grows bigger, and bigger, and bigger, without bound. The graph shoots up (or down) a forbidden vertical line and never comes back.

This is the second fundamental way a limit can misbehave:

• Finite limit — $f(x)\to L$, a nice real number. Example: $\frac{\sin x}{x}\to 1$.
• Infinite limit — $f(x)\to\pm\infty$. Example: $\frac{1}{x^{2}}\to+\infty$ as $x\to 0$.
• No limit at all — the function oscillates or the two sides disagree. Example: $\sin(1/x)$ near 0.

When we say "the limit is infinity" we are not claiming that the limit exists. Infinity is not a real number. We are recording the specific way the limit fails to exist — the function leaves every ceiling behind.

Intuition: Growing Without Bound

Picture sliding your finger along the x-axis toward a fixed point $a$. At each x you peek at the height of the graph. Normally those heights cluster around some value. Sometimes though, the heights don't cluster — they run away.

A cartoon. Imagine standing on a trampoline next to a column that reaches the ceiling. The closer you walk toward the column, the more the trampoline floor slopes up. One step before the column the floor is at your knees. A centimetre before the column the floor is above your head. A millimetre before, the floor pokes through the ceiling. That column is a vertical asymptote.

Concretely, for $f(x)=1/x$ near $x=0$:

No single number is the limit. For any ceiling you pick — $10$, $10^{6}$, $10^{100}$ — the function eventually pokes through it. That unbounded escape is what we mean by $f(x)\to+\infty$.

Interactive Exploration

Pick a function. Move the asymptote with the first slider. Then shrink $\varepsilon$ — the distance between the probe points and $a$ — and watch the two boxed values race off to infinity. The red dashed line marks the forbidden vertical:

Formal Definition of Infinite Limits

The informal idea "output grows without bound" is turned into hard mathematics by reversing the roles of $\varepsilon$ and $\delta$ from the ordinary limit definition. Instead of forcing the output into a band, we ask: for any target ceiling M, can we force the output above M by pushing x close enough to a?

Seeing It in Numbers

Formal statements are reassuring, but tables make the idea visceral. Pick a function and watch what happens as $h=|x-2|$ shrinks by a factor of 10 each row:

Worked Example 1 — A Simple Pole

Compute the one-sided limits of

as $x\to 2$, and decide whether the two-sided limit exists.

Four Flavors of Vertical Asymptote

Not every asymptote behaves the same way. By matching the left-hand and right-hand infinite limits, we get four canonical cases. Use the gallery below as a lookup table the next time you have to classify one:

Cases I and II are the two cases where mathematicians allow themselves to write $\lim=\pm\infty$. In Case III the two sides disagree, so the limit does not exist at all — we must either pick a side or accept no answer.

Vertical Asymptotes — Definition

A function $f$ has a vertical asymptote at $x=a$ if at least one of the one-sided limits there is infinite:

Notice the or: a single exploding one-sided limit is enough. The function $\ln x$ has a vertical asymptote at $x=0$ even though only the right side $0^{+}$ makes sense — it dives to $-\infty$.

How to Find Vertical Asymptotes

Three families of functions cover 95% of the cases you meet:

1. Rational functions

For $f(x)=p(x)/q(x)$, a vertical asymptote at $x=a$ happens when $q(a)=0$ and $p(a)\neq 0$. If both numerator and denominator vanish, the zero might be a removable hole instead — you must cancel common factors first.

2. Logarithms

$\ln(u(x))$ has a vertical asymptote wherever $u(x)=0^{+}$. The logarithm's left side is undefined, so usually only a one-sided limit applies.

3. Trigonometric functions

$\tan x,\; \sec x,\; \csc x,\; \cot x$ have vertical asymptotes at every zero of their inner cosine or sine: $\tan x$ blows up at $x=\tfrac{\pi}{2}+k\pi$.

Worked Example 2 — A Rational Function

Find all vertical asymptotes of

Python Implementation

Before reaching for a library, let's do the experiment by hand. Click any line below to see the actual value of every variable on that line — across all five iterations of the loop.

1# Plain Python: watch f(x) = 1 / (x - 2) as x approaches 2
2def f(x):
3    """Has a vertical asymptote at x = 2."""
4    return 1 / (x - 2)
5
6a = 2                                 # asymptote location
7step_sizes = [1, 0.1, 0.01, 0.001, 0.0001]
8
9print(f"{'h':>8}  {'x=a-h':>10}  {'f(a-h)':>14}  {'x=a+h':>10}  {'f(a+h)':>14}")
10for h in step_sizes:
11    x_left  = a - h                   # approach from below
12    x_right = a + h                   # approach from above
13    f_left  = f(x_left)               # → negative
14    f_right = f(x_right)              # → positive
15    print(f"{h:>8}  {x_left:>10.5f}  {f_left:>14.4f}  {x_right:>10.5f}  {f_right:>14.4f}")

PyTorch: Why Asymptotes Break Training

Vertical asymptotes are not just a calculus curiosity — they are a very real failure mode in machine learning. When a loss function or one of its components passes near a pole, the value of the loss explodes, and the gradient explodes even faster (by one extra power). The next snippet makes that concrete: compute $f(x)=1/(x-2)$ and its gradient with PyTorch's autograd, then watch both race to infinity together.

1import torch
2
3# Show both the value AND the gradient exploding near x = 2.
4a = 2.0
5
6# Tensor of x-values approaching 2 from the right; enable autograd.
7x = torch.tensor([2.1, 2.01, 2.001, 2.0001, 2.00001], requires_grad=True)
8
9# Forward pass: elementwise f(x) = 1 / (x - a)
10f = 1.0 / (x - a)
11
12# Backward pass: accumulates df/dx_i into x.grad.
13#   d/dx [1/(x-a)] = -1/(x-a)^2
14f.sum().backward()
15
16print(f"{'x':>10}   {'f(x)':>14}   {'|f\'(x)|':>14}")
17for xi, fi, gi in zip(x.tolist(), f.tolist(), x.grad.tolist()):
18    print(f"{xi:>10.5f}   {fi:>14.4f}   {abs(gi):>14.4g}")

Real-World Applications

Common Pitfalls

Summary

Key Takeaways

1. An infinite limit describes a specific way a limit fails to exist — the output grows past every ceiling.
2. The formal definition reverses the roles of $\varepsilon$ and $M$: pick any ceiling, we'll give you a tolerance.
3. Vertical asymptotes come in four flavors; the sign analysis determines which.
4. Rational functions: factor, cancel, check numerator — then read off the asymptotes.
5. In machine learning, a pole in the forward pass is a worse pole in the backward pass. Always clamp or reparametrize.

Coming next: §2.5 introduces the ε–δ definition — the same "name-a-ceiling" game we just played, but played for finite limits. You'll see that the infinite-limit definition was the easier sibling all along.