Limits

Definition

\[\displaystyle \lim_{x \to c}f(x) = L\]

Means that $f(x)$ approaches as close as possible to $L$ by making $x$ approach as close as possible to $c$, without being equal to $c$.

Rules

$\displaystyle \lim_{x\to a}c = c$

$\displaystyle \lim_{x\to a}x = a$

$\displaystyle \lim_{x\to a}x^n = a^n$

$\displaystyle \lim_{x\to a}x^\frac{1}{n} = a^\frac{1}{n}$

$\displaystyle \lim_{x\to 0}\frac{\sin x}{x} = 1$

$\displaystyle \lim_{x\to 0}\frac{x}{\sin x} = 1$

$\displaystyle \lim_{x\to 0}\frac{\cos x -1}{x} = 0$

$\displaystyle \lim_{x\to 0}\frac{x}{\cos x - 1} = 0$

$\displaystyle \lim_{x\to 0}\frac{\tan x}{x} = 1$

$\displaystyle \lim_{x\to 0}\frac{x}{\tan x} = 1$

$\displaystyle \lim_{x\to \infty}\frac{1}{x^r} = 0 \qquad \{r > 0, r \in \mathbb{Q}\}$

$\displaystyle \lim_{x\to -\infty}\frac{1}{x^r} = 0 \qquad \{r < 0, r \in \mathbb{Q}\}$

$\displaystyle \lim_{x\to -\infty}x^r = \infty$ (i.e. limit doesn’t exist)

General rules

Assume both $\displaystyle \lim_{x\to a}f(x)$ and $\displaystyle \lim_{x\to a}g(x)$ exist:

$\displaystyle \lim_{x\to a}f(x)\pm g(x) = \lim_{x\to a}f(x) \pm \lim_{x\to a}g(x)$

$\displaystyle \lim_{x\to a}c\cdot f(x) = c \cdot \lim_{x\to a}f(x)$

$\displaystyle \lim_{x\to a}f(x)\cdot g(x) = \lim_{x\to a}f(x) \cdot \lim_{x\to a}g(x)$

$\displaystyle \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\displaystyle \lim_{x\to a}f(x)}{\displaystyle \lim_{x\to a}g(x)} \qquad \{\lim g(x) \neq 0\}$

$\displaystyle \lim_{x\to a}f(x)^n = (\lim_{x\to a}f(x))^n$

$\displaystyle \lim_{x\to a}f(x)^\frac{1}{n} = (\lim_{x\to a}f(x))^\frac{1}{n}$

Solution approaches

Direct substitution

If a function is continuous at the point which the limit approaches, then the value can be directly substituted into the function.

Therefore, try to rewrite a function into a function that’s continuous at the point the limit is approaching.

Use one-sided limits

$\displaystyle\lim_{x\to a}f(x)$ exists if both $\displaystyle\lim_{x\to a-}f(x)$ and $\displaystyle\lim_{x\to a+}f(x)$ exist and are equal.

Useful to find the limit of discontinuous functions.

Pinching theorem

Attempt to find two functions $h$ and $g$ such that $h(x) \leq f(x) \leq g(x)$ and both function’s limits are equal at the point being approached by the limit being calculated. Then the limit of the two functions is equal to the limit being calculated.

Useful for functions where limit rules don’t apply, such as where a portion of the function’s limit doesn’t exist.

Rewrite limit as other known limits

For example, attempt to rewrite the limit as the definition of the derivative

\[f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} = \frac{f(b) - f(a)}{b - a}\]

L’Hospital’s rule

If both $\displaystyle\lim_{x\to a}f(x) = 0$ and $\displaystyle\lim_{x\to a}g(x) = 0$ or both are equal to $\pm \infty$ then

\[\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}\]

Possible results that can be rewritten so that l’Hospital’s rule applies, include those that at first sight might look like they approach:

\[\frac{\infty}{\infty}, \frac{0}{0}, 0\cdot\infty, \infty\cdot-\infty, 0^0, 1^\infty, \infty^0\]