Question

Suppose that ƒ is an odd function of x. Does knowing that limxs0+ ƒ(x) = 3 tell you anything about limxs0- ƒ(x)? Give reasons for your answer

Answer 1

we are given

f(x) is a odd function

so, $f(-x)=-f(x)$

$-f(x)=f(-x)$

now, we are given

$\lim _{x\to 0+}f\left(x\right)=3$

Let's multiply both sides by -1

$-\lim _{x\to 0+}f\left(x\right)=-3$

$\lim _{x\to 0+}-f\left(x\right)=-3$

we have

$-f(x)=f(-x)$

$\lim _{x\to 0+}f\left(-x\right)=-3$

now, let's consider y=-x

so, when x--->0+

but for y=-x

y--->0-

now, we can plug it

$\lim _{y\to 0-}f\left(y\right)=-3$

replace y as x

we get

$\lim _{x\to 0-}f\left(x\right)=-3$.................Answer