MATH FACILITY: CALCULUS

Tangent:
The word tangent is derived from the Latin word tangens, which means “touching.” Thus a tangent to a curve is a line that touches the curve.
Limit of a function:
Suppose f(x) is deﬁned when is near the number

$a$ . (This means that

$f$ is deﬁned on some open interval that contains

$a$ , except possibly at

$a$ itself.) Then we write

$\lim_{x\to a}f(x)=L$
and say the "the limit of

$f(x)$ as

$x$ approaches

$a$ ,equals L"

if we can make the values of

$f(x)$ arbitrarily close to L (as close to L as we like) by taking

$x$ to be sufﬁciently close to

$a$ see more

Definition: Let

$f$ be a function deﬁned on some open interval that contains the number

$a$ , except possibly at

$a$ itself. Then we say that the limit of

$f(x)$ as

$x$ approaches

$a$ is

$L$ , and we write

$\lim_{x\to a}{f(x)}=L$
if for every number

$\epsilon>0$ there is a number

$\delta >0$ such that

$if \:\:0<|x-a|<\delta \:\: then \:\: |f(x)-L|<\epsilon$

Since

$|x-a|$ is the distance from

$x$ to

$a$ and

$|f(x)-L|$ is the distance from

$f(x)$ to

$L$ , and since

$\epsilon$ can be arbitrarily small, the deﬁnition of a limit can be expressed in words as follows:

$\lim_{x\to a}{f(x)}=L$ means that the distance between f(x) and L can be made arbitrarily small by taking the distance see more

Definition: A function

$f$ is continuous at a number

$a$ if

$\lim_{x\to a}{f(x)}=f(a)$
Notice that Definition implicitly requires three things if

$f$ is continuous at

$a$
1. f(a) is defined (that is,

$a$ is in domain of

$f$ )
2.

$\lim_{x\to a}{f(x)}$ exists
3.

$\lim_{x\to a}{f(x)}=f(a)$
The deﬁnition says that

$f$ is continuous at

$a$ if

$f(x)$ approaches

$f(a)$ as x approaches a.
Right Continuity
A function

$f$ is continuous from right at a number

$a$ if

$\lim_{x\to a^{+}}{f(x)}=f(a)$
Left Continuity
A function

$f$ is continuous from left at a number see more