Limit Laws and Computation Strategies

Learning Objectives

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

1. State and apply the six limit laws (sum, difference, product, quotient, constant-multiple, power/root) to assemble new limits out of known ones.
2. Recognise when direct substitution is legal and when it produces an indeterminate form such as $0/0$.
3. Choose the right algebraic trick — factor, rationalise, common denominator, or trigonometric identity — to resolve a 0/0 limit.
4. Connect every limit law to the ε–δ definition from the previous section and understand why each law is a theorem, not a convention.
5. Use Python and PyTorch to cross-check symbolic answers numerically and to compute 0/0 limits via autograd.

Why We Need Laws, Not Just ε–δ

"The ε–δ definition tells us what a limit is. The limit laws tell us how to compute one without starting from scratch every time."

Imagine you need to evaluate $\displaystyle\lim_{x \to 3}\bigl(x^{2} + 3x + 1\bigr)$. You could run an ε–δ proof from scratch — pick any ε, chase δ through the algebra of a quadratic, verify the bound. That would take a page. Or you could argue:

• $\displaystyle\lim_{x \to 3} x^{2} = 9$ (known),
• $\displaystyle\lim_{x \to 3} 3x = 9$ (known),
• $\displaystyle\lim_{x \to 3} 1 = 1$ (trivial),
• so the whole thing equals $9 + 9 + 1 = 19$.

That second path is one line long. It works because limits distribute over addition, multiplication, and the other algebraic operations — provided we can prove they do. The proofs come straight from ε–δ (we give a pattern in the pitfalls section); the payoff is that we never have to write them again.

The Limit Laws

Suppose $\displaystyle\lim_{x \to c} f(x) = L$ and $\displaystyle\lim_{x \to c} g(x) = M$ both exist, and let k be any constant. Then each of the following limits exists and equals the stated value.

Interactive: Watch the Laws Combine

The picture below shows two reference curves $f(x) = x^{2} + 1$ and $g(x) = 2x + 1$, plus the combined curve produced by whatever law you pick. Drag the limit point c, and watch the combined curve's value at c match exactly what the right-hand side of each law predicts.

1. Pick the sum law. Slide c to 1. f(1) = 2, g(1) = 3, sum = 5. The predicted limit L = 5 shows up on the violet dot — that is exactly where the green curve crosses x = 1.
2. Switch to product. At c = 1: f · g = 2 · 3 = 6. The green curve's height at x = 1 should equal 6.
3. Try quotient at c = 1 — ratio 2/3 ≈ 0.667. Now slide c toward the point where g(c) = 0 (i.e. c = −0.5). The green curve blows up — a live demonstration that the quotient law needs $\lim g \neq 0$.
4. Shrink the distance |x − c| to 10⁻⁶ and watch the numerical approach table. The last row converges to the predicted L within float precision — that is the law at work.

Strategy 1 — Direct Substitution

The first thing to try, every time, is direct substitution: plug c into the formula and see what comes out. If the formula is continuous at c, you are done.

Example. $\displaystyle\lim_{x \to 3}\bigl(x^{2} + 3x + 1\bigr) = 3^{2} + 3(3) + 1 = 19$. Polynomials are continuous, so we plug in and finish.

Example. $\displaystyle\lim_{x \to 0}\dfrac{\cos x}{1 + x^{2}} = \dfrac{\cos 0}{1 + 0} = 1$. The denominator 1 + x² is never zero, so the quotient law applies; both pieces are continuous, so plug in.

When Substitution Fails: 0 / 0

The most common failure is $0/0$. Plugging in makes both numerator and denominator vanish simultaneously. This does not mean the limit is undefined — it means the quotient law's hypotheses are violated, so the law cannot be applied in that form. The fix: rewrite the expression so the offending factor cancels, then try substitution again.

Interactive: 0 / 0 Strategy Lab

Pick a 0/0 problem and step through the algebraic transformation. The plot shows the original function with a hollow circle at the discontinuity and its continuous extension in green — the same value at every x except the offending point.

1. Start with Factor & cancel. Step through the derivation. Notice that the blue dashed curve and the green solid curve are identical away from x = 2.
2. Switch to Rationalize. The original curve looks almost constant near x = 0, because the slope is 1/6 — a small number. Shrink the probe distance and watch both columns converge to 0.16666…
3. Try Common denominator. The numerator trick turns a composite fraction into a single clean rational expression.
4. Trig identity is the hardest — multiply by (1 + cos x) / (1 + cos x), use the Pythagorean identity, then pull out the famous limit sin x / x = 1 (which we will prove in the Squeeze-Theorem section).

Worked Example: Factor and Cancel

Compute $\displaystyle\lim_{x \to 2}\dfrac{x^{2} - 4}{x - 2}$.

A Decision Flowchart

Python: Verifying Limits Numerically

The laws give us the answer instantly; a short Python script gives us confidence that the answer is right. The pattern is always the same: evaluate the function at points that shrink toward c and watch the output stabilise.

1def f(x):
2    """Original form: (x^2 - 4) / (x - 2). Undefined at x = 2 (0/0)."""
3    return (x * x - 4) / (x - 2)
4
5def f_simplified(x):
6    """After factoring and cancelling (x-2): f(x) = x + 2 for x != 2."""
7    return x + 2
8
9# Numerical approach table — sample x on both sides of 2.
10c = 2
11offsets = [-0.1, -0.01, -0.001, -1e-6, 1e-6, 0.001, 0.01, 0.1]
12
13print(f"{'dx':>8}  {'x':>10}  {'f(x)':>14}")
14print("-" * 38)
15for dx in offsets:
16    x = c + dx
17    print(f"{dx:>+8.0e}  {x:>10.6f}  {f(x):>14.8f}")
18
19# Direct substitution into the *simplified* form is allowed
20# because x + 2 is a polynomial (continuous everywhere).
21limit_via_factor = f_simplified(c)
22print()
23print(f"lim_(x -> {c}) f(x)  =  {limit_via_factor}   (by factor-and-cancel)")

PyTorch: Limits as Derivatives

Here is a beautiful connection. The expression $\displaystyle\dfrac{x^{2} - 4}{x - 2}$ is the difference quotient for the function $f(x) = x^{2} - 4$ based at the point 2, because $f(2) = 0$, so $\dfrac{f(x) - f(2)}{x - 2} = \dfrac{x^{2} - 4}{x - 2}$. The limit of a difference quotient is, by definition, the derivative. So the answer to our 0/0 limit is $f'(2) = 2x\big|_{x=2} = 4$ — and PyTorch's autograd delivers that in one call.

1import torch
2
3# The indeterminate form (x^2 - 4) / (x - 2) at x = 2 has
4# the exact structure of a difference quotient with f(x) = x^2 - 4:
5#   lim_(x -> 2)  (f(x) - f(2)) / (x - 2)  =  f'(2)
6# So the "limit law" answer should equal the derivative at 2.
7
8def num(x):
9    return x * x - 4           # numerator, with num(2) = 0
10
11# 1) Numerical confirmation: shrink |x - 2| and watch the ratio.
12c = 2.0
13print("Numerical approach:")
14for delta in [1e-1, 1e-2, 1e-3, 1e-6, 1e-9]:
15    x = torch.tensor(c + delta, dtype=torch.float64)
16    ratio = num(x) / (x - c)
17    print(f"  x - c = {delta:>7.0e}   ratio = {ratio.item():.10f}")
18
19# 2) Symbolic answer via autograd: f'(2).
20x = torch.tensor(c, dtype=torch.float64, requires_grad=True)
21y = num(x)                     # y = 0 at c = 2, but the GRAPH is what matters
22y.backward()                   # fills x.grad with f'(c)
23
24print()
25print(f"f'(c = {c})  =  {x.grad.item():.10f}   <-- this IS the limit")

Pitfalls and Fine Print

Summary

The limit laws turn the ε–δ definition into a practical tool. Given known limits L and M, every algebraic combination has a predictable limit — as long as the hypotheses (especially M ≠ 0 for quotients and L ≥ 0 for even roots) are met. When substitution produces an indeterminate form, a handful of algebraic tricks almost always rewrites the expression into something the laws can finish.

Key Takeaways

1. The limit laws (sum, difference, product, quotient, constant-multiple, power/root) are theorems derived from ε–δ — not conventions. Each one has hypotheses that must be checked before use.
2. Direct substitution is the first move; it works for every continuous function at every point in its domain.
3. $0/0$ is not a dead end — it is a signal to rewrite. Factor, rationalize, common-denominator, and trig-identity handle the common cases.
4. The silver-bullet argument: two expressions that agree everywhere except at c share the same limit at c. Cancelling a common factor produces exactly such a pair.
5. Every $(f(x) - f(c))/(x - c)$ limit is a derivative — limit laws and autograd are two views of the same underlying operation.

Coming Next: Some limits refuse to yield to the laws alone — they oscillate, or they hide between bounds that only a clever pair of inequalities can trap. The next section introduces the Squeeze Theorem, the tool that handles the stubborn cases and delivers one of the most famous limits in calculus: $\lim_{x \to 0} \sin(x)/x = 1$.