The limits of sin x over x and related trig limits, the limit that defines e, the definition of continuity, types of discontinuity, and the Intermediate and Extreme Value Theorems.
Two special trig limits
The first one is proved geometrically with the squeeze theorem: on the unit circle, \(\cos x \le \frac{\sin x}{x} \le 1\) for small \(x\), and both outer functions approach \(1\). It says that for small angles in radians, \(\sin x \approx x\).
To use it, make the argument of sine match the denominator.
The limit that defines \(e\)
\[\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n = e \approx 2.71828\]
This is the limit behind continuously compounded interest.
Continuity
Informally, you can draw the graph near \(c\) without lifting your pencil. Polynomials, rational functions, roots, exponentials, logs and trig functions are continuous on their domains, which is exactly why direct substitution works for them.
Types of discontinuity
Type
What happens
Example
removable (a hole)
the limit exists, but \(f(c)\) is missing or different
\(\frac{x^2 - 4}{x - 2}\) at \(x = 2\)
jump
one-sided limits exist but differ
a step function
infinite
the function blows up
\(\frac{1}{x}\) at \(x = 0\)
Two big theorems
In particular, if \(f(a)\) and \(f(b)\) have opposite signs, \(f\) has a root between \(a\) and \(b\).
Practice
Quick check
What is \(\lim_{x \to 0} \frac{\sin x}{x}\)?
This is the fundamental trig limit.
Quick check
\(f\) has a removable discontinuity at \(c\). Which statement is true?
A removable discontinuity is a hole: the limit exists but doesn't match \(f(c)\).
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