GRE Mathematics Subject Test

## Limits The statement $\lim_{x \to a} f(x) = L$ means that $f(x)$ can be made as close to $L$ as we like by taking $x$ close enough to $a$, without $x$ being equal to $a$. The function need not be defined at $a$, and even if it is, its value there plays no part in the limit. In the formal version, for every $\varepsilon > 0$ there is a $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \varepsilon$. A **one-sided limit** approaches $a$ from one side only, written $\lim_{x \to a^-}$ and $\lim_{x \to a^+}$. The two-sided limit exists exactly when both one-sided limits exist and are equal. For $f(x) = |x|/x$ the left limit at $0$ is $-1$ and the right limit is $1$, so there is no limit at $0$. A limit at infinity describes behaviour as $x$ grows without bound, and an infinite limit means the values grow without bound. For a rational function, divide numerator and denominator by the highest power of $x$ in the denominator. For example $$ \lim_{x \to \infty} \frac{3x^2 - x + 5}{2x^2 + 7} = \lim_{x \to \infty} \frac{3 - 1/x + 5/x^2}{2 + 7/x^2} = \frac{3}{2}. $$ ### Limit laws If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$ both exist, then the limit of a sum, difference, product and constant multiple is the corresponding combination of $L$ and $M$, the limit of a quotient is $L/M$ provided $M \neq 0$, and $\lim f(x)^n = L^n$. If $f$ is continuous at $M$ then $\lim f(g(x)) = f(M)$. ### The squeeze theorem If $g(x) \le f(x) \le h(x)$ near $a$ (except…