Unit 1: Limits and Continuity

17 terms. Say what each one means, then open the card to check.

• Limit

  The number $f(x)$ gets as close to as you like when $x$ is close enough to $a$ on both sides. The value $f(a)$ itself plays no part.

  Example. If $f(x)$ is 4.9, 4.99, 5.01 and 5.1 at inputs closing in on 2 from both sides, a table suggests $\lim_{x \to 2} f(x) = 5$, whatever $f(2)$ is.

• One-sided limit

  The number $f(x)$ approaches as $x$ nears $a$ from one side only: $\lim_{x \to a^-}$ uses inputs below $a$ and $\lim_{x \to a^+}$ uses inputs above $a$.

  Example. If a graph runs at height 1 just left of $x = 0$ and at height 3 just right of it, the left-hand limit at 0 is 1 and the right-hand limit is 3.

• A limit that does not exist

  A two-sided limit fails to exist when the one-sided limits differ, when the outputs grow without bound, or when they keep swinging without settling.

  Example. $\sin\left(\frac{1}{x}\right)$ swings between -1 and 1 infinitely often as $x$ nears 0, so its limit at 0 does not exist.

• Direct substitution

  For a function that is continuous at $a$, such as a polynomial, the limit at $a$ is simply the value of the function at $a$.

  Example. $\lim_{x \to 3}(x^2 + 1) = 3^2 + 1 = 10$.

• Indeterminate form $\frac{0}{0}$

  What you get when substituting into a quotient makes the top and the bottom both 0. It tells you nothing about the limit yet: rewrite the expression and substitute again.

  Example. $\frac{x^2 - 16}{x - 4}$ gives $\frac{0}{0}$ at 4, but it equals $x + 4$ for $x \ne 4$, so its limit at 4 is 8.

• Rationalizing

  Multiplying the top and bottom of a quotient by the conjugate of a square-root expression, to clear a $\frac{0}{0}$ form.

  Example. For $x \ne 25$, $\frac{\sqrt{x} - 5}{x - 25} = \frac{1}{\sqrt{x} + 5}$, so the limit as $x \to 25$ is $\frac{1}{10}$.

• Squeeze theorem

  If $g(x) \le f(x) \le h(x)$ for $x$ near $a$, and $g$ and $h$ both have limit $L$ at $a$, then $f$ also has limit $L$ at $a$.

  Example. Near 2, $-(x-2)^2 \le (x-2)^2\cos\left(\frac{1}{x-2}\right) \le (x-2)^2$, and both bounds go to 0, so the middle function has limit 0 at 2.

• The limit of $\frac{\sin x}{x}$

  $\lim_{x \to 0}\frac{\sin x}{x} = 1$ when $x$ is in radians. Many trig limits reduce to it.

  Example. $\frac{\tan x}{x} = \frac{\sin x}{x} \cdot \frac{1}{\cos x}$, so $\lim_{x \to 0}\frac{\tan x}{x} = 1 \cdot 1 = 1$.

• Continuous at a point

  $f$ is continuous at $a$ when $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and the two are equal.

  Example. If $g(1) = 6$ and $g(x)$ approaches 6 from both sides of 1, then $g$ is continuous at 1.

• Removable discontinuity

  A point where the limit exists but the function value is missing or different from it. On a graph it is a hole, and redefining one value fixes it.

  Example. $\frac{x^2 - x}{x}$ is undefined at 0 but has limit -1 there, so defining its value at 0 as -1 removes the gap.

• Jump discontinuity

  A point where both one-sided limits exist but are different numbers, so the graph jumps from one height to another.

  Example. A parking fee that is 5 dollars for stays up to one hour and 8 dollars for longer stays has a jump at one hour.

• Infinite discontinuity

  A point where the function grows without bound on at least one side, so its graph has a vertical asymptote there.

  Example. $\frac{1}{(x+2)^2}$ has an infinite discontinuity at $x = -2$: its outputs pass every height as $x$ nears -2.

• Intermediate Value Theorem

  If $f$ is continuous on the closed interval $[a, b]$ and $N$ is strictly between $f(a)$ and $f(b)$, then $f(c) = N$ for at least one $c$ strictly between $a$ and $b$.

  Example. A temperature that changes continuously from 12 degrees at 6 a.m. to 20 degrees at noon must equal 15 degrees at some moment in between.

• Infinite limit

  A statement such as $\lim_{x \to a} f(x) = \infty$: the outputs grow without bound near $a$. The limit still does not exist as a number; the symbol says how it fails.

  Example. $\lim_{x \to 0}\frac{1}{x^2} = \infty$, since $\frac{1}{x^2}$ is above 100 whenever $0 < |x| < 0.1$ and keeps growing.

• Vertical asymptote

  The line $x = a$ is a vertical asymptote when $f(x)$ grows without bound, toward $\infty$ or $-\infty$, as $x$ approaches $a$ from at least one side.

  Example. $\lim_{x \to -4^+}\frac{1}{x + 4} = \infty$, so $x = -4$ is a vertical asymptote of $y = \frac{1}{x + 4}$.

• Limit at infinity

  The number $f(x)$ approaches as $x$ grows without bound ($x \to \infty$) or decreases without bound ($x \to -\infty$).

  Example. As $x \to \infty$, $\frac{5}{x}$ shrinks toward 0, so $\lim_{x \to \infty}\frac{5}{x} = 0$.

• Horizontal asymptote

  The line $y = L$ is a horizontal asymptote when $f(x)$ approaches $L$ as $x \to \infty$ or as $x \to -\infty$. It describes the ends of the graph only.

  Example. $\lim_{x \to \infty}\frac{10x - 1}{4x + 7} = \frac{10}{4} = 2.5$, so $y = 2.5$ is a horizontal asymptote.