Question

Show that limit x tends to 0 cos 1/x doesn't exist

Answer 1

Answer:

1. Please mark this answer as a brainlist
2. plz plz plz follow me
3. please thank me if the answer is helpful to you

Step-by-step explanation:

When x tends to 0, 1/x tends to infinity. Cos(1/x) does not tends to attain a limiting value,rather it oscillates b/w -1 and 1. So Limit Does Not Exists.

Answer 2

Limit

A function f(x) is said to have the limit l at a point x=a if x closed to a whenever f(x) approaches to l . We can write _(x→a)^lim f(x)=l

It means the difference between f(x) and l i.e. , |f(x)-l| is very small when the difference between x and a i.e. , |x-a| is very small.

In other words, a function f(x) is said to have the limit l at a point x=a if for given a small positive number ε , however small, there exists another positive number δ such that 0<|x-a|<δ whenever 0<|f(x)-l|<ε

This means to say that when x approaches a within a distance of δ , f(x) approaches l within a distance of ε .

Right hand limit

A function f(x) is said to have right hand limit l as x approaches a if and only if given any ε>0 , however small , there exists a positive number δ such that a<x<a+ δ ⇒ 0<|f(x)-l|<ε

In symbol , we write _(x→a^+)^lim f(x)=l or , _(x→a^(+0))^lim f(x)=l or , _(h→0)^lim f(a+h)=l or , f(a+0)=l

Left hand limit

A function f(x) is said to have a left hand limit l as x approaches a if and only if given any ε>0 , however small , there exists a positive number δ such that a- δ<x<a ⇒ 0<|f(x)-l|<ε

In symbol , we write _(x→a^-)^lim f(x)=l or , _(x→a^(-0))^lim f(x)=l or , _(h→0)^lim f(a-h)=l or , f(a-0)=l

It is to be noted that x→a^+ means a<x<a+ δ and x→a^- means a- δ<x<a but x≠a .

Necessary and Sufficient Condition for f(x) to have a limit

The limits _(x→a^+)^lim f(x) and _(x→a^-)^lim f(x) are respectively called right hand limit and left hand limit of f(x) at x=a .

Thus we can also say that a function f(x) has a limit at x=a , if and only if the right hand limit _(x→a^+)^lim f(x) is equal to the left hand limit _(x→a^-)^lim f(x) .

For example

Show that _(x→0)^lim sin⁡〖1/x〗 does not exist.

Show that _(x→0)^lim cos⁡〖1/x〗 does not exist.