what is limits and how are functions and limits related to each other
Answers
The limit is a thing that you have not to cross
Step-by-step explanation:
You have don't watch TV more than 30 HOURS
Answer:
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, a limit of a function is usually written as
lim x → c f ( x ) = L , {\displaystyle \lim _{x\to c}f(x)=L,} {\displaystyle \lim _{x\to c}f(x)=L,}
and is read as "the limit of f of x as x approaches c equals L". The fact that a function f approaches the limit L as x approaches c is sometimes denoted by a right arrow (→), as in
f ( x ) → L as x → c . {\displaystyle f(x)\to L{\text{ as }}x\to c.} {\displaystyle f(x)\to L{\text{ as }}x\to c.}
Suppose f is a real-valued function and c is a real number. Intuitively speaking, the expression
lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} \lim _{x\to c}f(x)=L
means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, the above equation can be read as "the limit of f of x, as x approaches c, is L".
Augustin-Louis Cauchy in 1821,[2] followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The definition uses ε (the lowercase Greek letter epsilon) to represent any small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (L − ε, L + ε), which can also be written using the absolute value sign as |f(x) − L| < ε.[2] The phrase "as x approaches c" then indicates that we refer to values of x whose distance from c is less than some positive number δ (the lower case Greek letter delta)—that is, values of x within either (c − δ, c) or (c, c + δ), which can be expressed with 0 < |x − c| < δ. The first inequality means that the distance between x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c.[2]
The above definition of a limit is true even if f(c) ≠ L. Indeed, the function f need not even be defined at c.
For example, if
f ( x ) = x 2 − 1 x − 1 {\displaystyle f(x)={\frac {x^{2}-1}{x-1}}} f(x)={\frac {x^{2}-1}{x-1}}
then f(1) is not defined (see indeterminate forms), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2:
f(0.9) f(0.99) f(0.999) f(1.0) f(1.001) f(1.01) f(1.1)
1.900 1.990 1.999 undefined 2.001 2.010 2.100
Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.
In other words, lim x → 1 x 2 − 1 x − 1 = 2 {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2} \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2
This can also be calculated algebraically, as x 2 − 1 x − 1 = ( x + 1 ) ( x − 1 ) x − 1 = x + 1 {\displaystyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1} {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1 for all real numbers x ≠ 1.
Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus lim x → 1 x 2 − 1 x − 1 = 1 + 1 = 2 {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=1+1=2} \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=1+1=2.
In addition to limits at finite values, functions can also have limits at infinity. For example, consider
f ( x ) = 2 x − 1 x {\displaystyle f(x)={2x-1 \over x}} f(x)={2x-1 \over x}
f(100) = 1.9900
f(1000) = 1.9990
f(10000) = 1.99990
Step-by-step explanation: