# 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
$$\quad \quad \quad \displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)} = \displaystyle \lim_{x\to a} \dfrac{f'(x)}{g'(x)}$$