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