Limits and discontinuities

What a limit is

• A limit describes what a function approaches as $x$ gets close to a value — not necessarily what it equals there.

$$\lim_{x \to a} f(x) = L$$

• Read this as: "as $x$ approaches $a$, $f(x)$ approaches $L$".
• The function need not be defined at $a$ at all. The limit is about the journey, not the destination.

Finding a limit by substitution

• For most functions, substitute the value in:

$$\lim_{x \to 3} (x^2 + 2x) = 9 + 6 = 15$$

• If substitution gives a number, that is the limit and you are finished.

When substitution gives $\frac{0}{0}$

• $\frac{0}{0}$ is called an indeterminate form. It does not mean the limit fails to exist — it means substitution was the wrong tool.
• Factorise and cancel, then substitute again:

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x - 2} = \lim_{x \to 2}(x + 2) = 4$$

• Note that $\frac{x^2-4}{x-2}$ is genuinely undefined at $x = 2$ — you cannot divide by zero. But the limit is still 4, because the function approaches 4 from both sides.

One-sided limits

• $\displaystyle\lim_{x \to a^-} f(x)$ is the left-hand limit — the value approached coming from below.
• $\displaystyle\lim_{x \to a^+} f(x)$ is the right-hand limit — approached from above.
• The two-sided limit exists only if both one-sided limits exist and are equal:

$$\lim_{x \to a} f(x) = L \quad \Longleftrightarrow \quad \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

• This is the test you apply to piecewise functions, which is how NZQA usually assesses limits.

The three kinds of discontinuity

• Removable discontinuity — a single hole. The limit exists, but the function is either undefined there or defined at the wrong value. Drawn as an open circle.
• Jump discontinuity — the left and right limits both exist but are different, so the two-sided limit does not exist. Typical of piecewise functions with mismatched pieces.
• Infinite discontinuity — the function grows without bound, giving a vertical asymptote. The limit does not exist.

Limits at infinity

• $\displaystyle\lim_{x \to \infty} f(x)$ describes the long-run behaviour — the horizontal asymptote, if there is one.

• For a rational function, compare the degrees of numerator and denominator:

  • Degree of top < degree of bottom — the limit is 0
  • Degrees equal — the limit is the ratio of the leading coefficients
  • Degree of top > degree of bottom — no finite limit; the function grows without bound

• Worked through:

  • $\displaystyle\lim_{x \to \infty}\frac{3x^2 + 1}{5x^2 - x} = \frac{3}{5}$ (equal degrees, leading coefficients 3 and 5)
  • $\displaystyle\lim_{x \to \infty}\frac{4x}{x^2 + 7} = 0$ (bottom wins)

• Useful limits worth knowing:

  • $$\displaystyle\lim_{x \to \infty} e^{-x} = 0$$
  • $$\displaystyle\lim_{x \to \infty} \frac{1}{x} = 0$$
  • $$\displaystyle\lim_{x \to 0^+} \ln x = -\infty$$

Vertical asymptotes

• A vertical asymptote occurs where the denominator is zero and the numerator is not.
• For $f(x) = \dfrac{x+1}{x-3}$, the denominator vanishes at $x = 3$ while the numerator gives 4, so there is a vertical asymptote at $x = 3$.
• If both vanish, factorise first — you may have a hole rather than an asymptote.