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Math Analysis 1

Limits of functions: definitions, theorems, indeterminate forms and asymptotes

Epsilon-delta definition, one-sided limits, calculation rules and the squeeze theorem, indeterminate forms and how to solve them, asymptotes.

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The limit of a function describes how f(x) behaves as x approaches a point, without caring about what happens at the point itself. It is the concept on which continuity, derivatives and integrals are built. This lesson gives the definitions, the rules of calculation and the methods for indeterminate forms.

The definitions

Let x_0 be an accumulation point of the domain of f. We write \lim_{x\to x_0} f(x) = L if

    \[ \forall \varepsilon > 0\ \exists \delta > 0 : \ 0 < |x - x_0| < \delta \Rightarrow |f(x) - L| < \varepsilon. \]

The condition 0 < |x - x_0| excludes the point x_0: the value f(x_0), if it exists, does not matter. The other variants come from changing the neighbourhoods: for x \to +\infty we ask x > M instead of |x - x_0| < \delta; for an infinite limit we ask f(x) > K instead of |f(x) - L| < \varepsilon.

One-sided limits. Approaching only from the right (x > x_0) or only from the left gives \lim_{x\to x_0^+} and \lim_{x\to x_0^-}. The limit exists if and only if both exist and are equal. For f(x) = \frac{|x|}{x} at 0 they are +1 and -1: the limit does not exist.

Rules of calculation

  • Uniqueness of the limit and sign preservation, as for sequences.
  • Algebra of limits for sums, products and quotients.
  • Squeeze: if g \le f \le h near x_0 and g, h \to L, then f \to L. This is how to prove x\sin\frac1x \to 0 as x \to 0: -|x| \le x\sin\frac1x \le |x|.
  • Composite functions and change of variable: if g(x) \to L and f is continuous at L, then f(g(x)) \to f(L) (see the in-depth article).
  • Link with sequences: f(x) \to L if and only if f(x_n) \to L for every sequence x_n \to x_0 with x_n \ne x_0. It is mostly used to show that a limit does not exist: \sin\frac1x as x \to 0 is 0 along x_n = \frac{1}{n\pi} and 1 along x_n = \frac{1}{\pi/2 + 2n\pi}.

Indeterminate forms and how to handle them

FormTypical techniqueExample
\frac{0}{0} with polynomialsfactor and cancel\frac{x^2-4}{x-2} = x + 2 \to 4 as x\to2
\frac{\infty}{\infty} with polynomialsfactor out the highest power\frac{2x^3+x}{5x^3-1} \to \frac25
\infty - \infty with rootsrationalise\sqrt{x^2+x} - x \to \frac12
\frac{0}{0} with transcendental functionsstandard limits and equivalences\frac{\sin 5x}{x} \to 5
1^\infty, 0^0, \infty^0write f^g = e^{g \ln f}x^x = e^{x\ln x} \to 1 as x\to0^+

For the example with the root: \sqrt{x^2+x} - x = \dfrac{x}{\sqrt{x^2+x} + x} = \dfrac{1}{\sqrt{1 + 1/x} + 1} \to \frac12. Later in the course l’Hôpital’s rule and Taylor expansions give even more general tools.

Asymptotes

Limits describe the graph “at the edges”:

  • vertical x = x_0 if f(x) \to \pm\infty as x \to x_0 (even from one side only);
  • horizontal y = L if f(x) \to L as x \to \pm\infty;
  • oblique y = mx + q if m = \lim \frac{f(x)}{x} and q = \lim (f(x) - mx) are finite, with m \ne 0.

Example: f(x) = \frac{x^2 + 1}{x - 1} has the vertical asymptote x = 1 and, since \frac{f(x)}{x} \to 1 and f(x) - x = \frac{x + 1}{x - 1} \to 1, the oblique asymptote y = x + 1.

A physical reading

The steady-state value of a circuit is a limit as t \to +\infty: the voltage on a charging capacitor, v(t) = V(1 - e^{-t/\tau}), tends to V, the horizontal asymptote. The gain of an amplifier with a very high open-loop gain, \frac{A}{1 + A\beta}, tends to \frac{1}{\beta} as A \to \infty: that is why feedback makes the gain independent of the transistors.

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