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