# Formulas, Rules and Theorems of Limits of Functions



## Definition

1) Formal Definition ("epsilon-delta definition")
The limit of $f(x)$, as $x$ approaches $a$ , exists and is equal to $L$ written as $\displaystyle \lim_{x\to a} f(x) = L$ if for any value of $\epsilon \gt 0$ we can find a value of $\delta \gt 0$such that if $0 \lt |x - a| \lt \delta$ then $|f(x) - L| \lt \epsilon$
2) Working Definition
The limit of $f(x)$, as $x$ approaches $a$ , exists and is equal to $L$ written as $\displaystyle \lim_{x\to a} f(x) = L$ if we can make the values of $f(x)$ as close as we want to $L$ as $x$ takes values closer to and on either sides of $a$.
Note that the function may or may not be defined at $x = a$ for a limit of a function to exist at $x = a$.

## Formulas of Limits

• $\displaystyle \lim_{x\to 0} \dfrac{\sin x}{x} = 1$ (See Squeezing Theorem to Find Limits of Mathematical Functions for proof)
• $\displaystyle \lim_{x\to 0} \dfrac{\sin^{-1} x}{x} = 1$
• $\displaystyle \lim_{x\to 0} \dfrac{\tan x}{x} = 1$
• $\displaystyle \lim_{x\to 0} \dfrac{\tan^{-1} x}{x} = 1$
• $\displaystyle \lim_{x\to 0} \dfrac{1 - \cos x}{x} = 0$ (See Limits of Trigonometric Functions for proof)
• $\displaystyle \lim_{x\to 0} \dfrac{a^x - 1}{x} = \ln a$ , if $a \gt 0$
• $\displaystyle \lim_{x\to 0} \dfrac{e^x - 1}{x} = 1$
• $\displaystyle \lim_{x\to \infty} (1+\dfrac{1}{x})^x = e$ (See Euler Constant e for definition of e)
• $\displaystyle \lim_{x\to \infty} (1+\dfrac{1}{x})^{kx} = e^k$
• $\displaystyle \lim_{x\to 0} (1+x)^{1/x} = e$
• $\displaystyle \lim_{x\to 0} \dfrac{\ln(1+x)}{x} = 1$
• $\displaystyle \lim_{x\to 0^+} \log_b x = -\infty$ , for $b \gt 1$
• $\displaystyle \lim_{x\to \infty} \log_b x = \infty$ , for $b \gt 1$
• $\displaystyle \lim_{x\to 0^+} \log_b x = \infty$ , for $0 \lt b \lt 1$
• $\displaystyle \lim_{x\to \infty} \log_b x = -\infty$ , for $0 \lt b \lt 1$
• $\displaystyle \lim_{x\to +\infty} b^x = \infty$ , for $b \gt 1$
• $\displaystyle \lim_{x\to - \infty} b^x = 0$ , for $b \gt 1$
• $\displaystyle \lim_{x\to +\infty} b^x = 0$ , for $0 \lt b \lt 1$
• $\displaystyle \lim_{x\to - \infty} b^x = \infty$ , for $0 \lt b \lt 1$
• $\displaystyle \lim_{x\to + \infty} x^{\frac{1}{x}} = 1$
Note that many of the limits of ratios listed above, can be found using the L'Hôpital's Rule.

## Theorems and Rules of Limits

Let $\displaystyle \lim_{x\to a} f(x) = L_1$ and $\displaystyle \lim_{x\to a} g(x) = L_2$
• $\displaystyle \lim_{x\to a} (f(x) \pm g(x)) = \displaystyle \lim_{x\to a} f(x) \pm \displaystyle \lim_{x\to a} g(x) = L_1 \pm L_2$
• $\displaystyle \lim_{x\to a} (f(x) \cdot g(x)) = \displaystyle \lim_{x\to a} f(x) \cdot \displaystyle \lim_{x\to a} g(x) = L_1 \cdot L_2$
• $\displaystyle \lim_{x\to a} (k \cdot g(x)) = k \cdot \displaystyle \lim_{x\to a} f(x) = k \cdot L_1$ , where $k$ is a constant
• $\displaystyle \lim_{x\to a} (\dfrac{f(x)}{g(x)}) = \dfrac{ \displaystyle \lim_{x\to a} f(x)}{ \displaystyle \lim_{x\to a} g(x) } = \dfrac{L_1}{L_2}$ if $L_2 \ne 0$
• $\displaystyle \lim_{x\to a} [f(x)]^n = [ \displaystyle \lim_{x\to a} f(x) ]^n = L_1^n$ , if $n$ is a positive integer.
• $\displaystyle \lim_{x\to a} \sqrt[n]{f(x)} = \sqrt[n] { \displaystyle \lim_{x\to a} {f(x)}} = \sqrt[n]{L_1}$ ; if $n$ is a positive integer and $L_1 \gt 0$ If $n$ even ,
• If $f(x)$ is a continuous function, then $\displaystyle \lim_{x\to a} f(x) = f(a)$

• Squeeze Theorem (or Sandwich Theorem)
If $f(x) \le g(x) \le h(x)$ for $x$ near $a$ and if $\displaystyle \lim_{x\to a} f(x) = \displaystyle \lim_{x\to a} h(x) = L$ , then $\displaystyle \lim_{x\to a} g(x) = a$

• L'Hôpital's Rule
If $\displaystyle \lim_{x\to a} f(x) = 0$ and $\displaystyle \lim_{x\to a} g(x) = 0$
or $\displaystyle \lim_{x\to a} f(x) = +\infty$ and $\displaystyle \lim_{x\to a} g(x) = +\infty$
or $\displaystyle \lim_{x\to a} f(x) = -\infty$ and $\displaystyle \lim_{x\to a} g(x) = -\infty$
and if $\displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)}$ exists , then
$\quad \quad \quad \displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)} = \displaystyle \lim_{x\to a} \dfrac{f'(x)}{g'(x)}$