Limits and Convergence

Part 2, Chapter 2: Single-Variable Real Analysis

Learning objectives

State the $\epsilon\text{-}\delta$ definition of a limit and read it as a challenge-response game
Prove simple limits (linear, quadratic, constant) using the $\epsilon\text{-}\delta$ machinery
Recognise why a single function can have different one-sided limits or no limit at all
Identify where rigorous limits appear in numerical analysis, signal processing, and physics

The limit is the foundational object of real analysis. Newton and Leibniz built calculus in the 1670s on an intuition of "infinitely small" quantities, but for nearly two centuries the logical foundation was a mess: tangent lines were slopes between two points that were "almost" the same point, and integrals were sums of rectangles with "infinitesimal" width. Cauchy (1820s) and Weierstrass (1850s) finally replaced this mysticism with a precise definition built entirely from inequalities. Once you understand the $\epsilon\text{-}\delta$ definition, every later concept in analysis, continuity, the derivative, the Riemann integral, uniform convergence, becomes a clean variation on the same theme.

The $\epsilon\text{-}\delta$ definition

We say $\lim_{x\to a}f(x)=L$ xtoaf(x)=L if for every $\epsilon>0$ there exists $\delta>0$ such that whenever $0<|x-a|<\delta$ , we have $|f(x)-L|<\epsilon$ . Read this as a challenge-response game: an opponent picks any tolerance $\epsilon$ (no matter how tiny), and you must respond with a window $\delta$ around $a$ that keeps $f(x)$ within $\epsilon$ of $L$ . If you can always win, the limit exists and equals $L$ .

Notice three subtleties: (i) the input is required to satisfy $0<|x-a|$ , so the value $f(a)$ itself is irrelevant, we only care about the behaviour near $a$ . (ii) The order of quantifiers matters: $\delta$ may depend on $\epsilon$ but not on $x$ . (iii) The conclusion is a strict inequality, but the proof technique still works with $\leq$ .

A worked proof: $\lim_{x\to 3}(2x+1)=7$ xto3(2x+1)=7

Given $\epsilon>0$ , choose $\delta=\epsilon/2$ . Then $|x-3|<\delta$ implies $|2x+1-7|=|2x-6|=2|x-3|<2\delta=\epsilon$ . The key trick: we computed $|f(x)-L|$ in terms of $|x-a|$ first, then solved for the $\delta$ that forces this expression below $\epsilon$ .

One-sided limits and non-existence

The two-sided limit $\lim_{x\to a}f(x)$ xtoaf(x) exists only if both one-sided limits $\lim_{x\to a^-}f$ xtoa−f and $\lim_{x\to a^+}f$ xtoa+f exist and are equal. The function $f(x)=|x|/x$ has $\lim_{x\to 0^-}=-1$ xto0−=−1 and $\lim_{x\to 0^+}=+1$ xto0+=+1, so $\lim_{x\to 0}f$ xto0f does not exist. The function $\sin(1/x)$ near $x=0$ fails even worse, it has no one-sided limit because it oscillates infinitely fast.

Use the function-grapher above to plot $f(x)=(x^2-1)/(x-1)$ near $x=1$ . The graph looks like the line $y=x+1$ with a hole at the point $(1,2)$ . Even though $f(1)$ is undefined (division by zero), the limit $\lim_{x\to 1}f(x)=2$ xto1f(x)=2 exists because the $\epsilon\text{-}\delta$ definition only cares about $0<|x-1|$ . Try also plotting $\sin(1/x)$ on a tiny window like $[-0.1, 0.1]$ to see why that limit does not exist.

Where this shows up

Numerical analysis & floating-point error bounds: Every theorem about how close a finite-precision answer is to the exact answer is an $\epsilon\text{-}\delta$ statement, "to get error below $\epsilon$ , use mesh size $\delta$ ." IEEE 754 error analysis, condition numbers, and convergence proofs for iterative solvers (Newton, conjugate gradient) all reduce to limit definitions.
Signal processing & Fourier analysis: Convolutions, low-pass filters, and frequency-response calculations all assume signals are continuous, which is itself an $\epsilon\text{-}\delta$ statement. Sampling theorems (Nyquist-Shannon) hinge on limits of sinc series.
Physics & instantaneous quantities: Instantaneous velocity, acceleration, current, and power are all defined as limits of average quantities, the $\epsilon\text{-}\delta$ idea is what makes Newton's second law $F=ma$ unambiguous at a single instant.

Pause and think: Why does the definition use $0<|x-a|<\delta$ instead of $|x-a|<\delta$ ? Construct a function where these two phrasings would disagree about whether the limit exists.

Try it

Predict first: for $\lim_{x\to 5}(4x-3)=17$ xto5(4x−3)=17, which $\delta$ works in terms of $\epsilon$ ? Then verify by computing $|4x-3-17|$ explicitly.
For $f(x)=x^2$ near $x=2$ with $\epsilon=0.01$ , find a concrete numerical $\delta$ . Hint: factor $|x^2-4|=|x-2||x+2|$ and bound $|x+2|$ on a small interval.
True or false: if $\lim_{x\to a}f(x)=L$ xtoaf(x)=L exists, then $f(a)$ must equal $L$ . Justify with one sentence and an example.
Show that $\lim_{x\to 0}\sin(1/x)$ xto0sin(1/x) does not exist by exhibiting two sequences $x_n,y_n\to 0$ along which $f(x_n)\to 0$ n)to0 but $f(y_n)\to 1$ n)to1.

A trap to watch for

Beginners often pick $\delta$ that depends on $x$ . That is illegal: $\delta$ must be chosen once for the given $\epsilon$ , before $x$ is revealed. Concretely, in a quadratic-limit proof you may need a preliminary bound like "first require $|x-a|<1$ ", then choose $\delta=\min(1,\epsilon/M)$ where $M$ is the supremum of the relevant factor over that pre-bounded region. The $\min$ trick is the universal escape hatch for non-linear functions.

What you now know

You can state the $\epsilon\text{-}\delta$ definition, prove linear and simple non-linear limits, diagnose failed limits via one-sided limits or oscillating subsequences, and recognise the same machinery in error analysis and physics. The next section §2.2 builds continuity directly on this foundation: a function is continuous at $a$ precisely when its limit at $a$ equals its value $f(a)$ .

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 2.
Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 4.
Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer, ch. 4.
Bartle, R. G., Sherbert, D. R. (2011). Introduction to Real Analysis (4th ed.). Wiley, ch. 4.
Spivak, M. (2008). Calculus (4th ed.). Publish or Perish, ch. 5.