Question

dicuss continuity f(x)=sinx x for all reall number

Answer 1

«Solution»

f(x) = sinx

Let's use some basic facts ...

lim Sinx = 0

x-->0

it is already clear from the graph of sinx near 0.

Now, observe that f(x) = sinx is defined for every real number. Let c be a real number.

Put x = c + h. If x ----> c we know that h---> 0.

Therefore

lim f(x) = lim sinx

x-->c. x--->c

=> lim sin(c + h)

h--->0

=> lim [sinccosh + coscsinh]

h--->0

=> lim [sinccosh] + lim[coscsinh]

h-->0. h-->0

=> sinc + 0 = sinc

sinc = f(c)

Thus, lim f(x) = f(c) and hence f is a continuous function.