Question

If f is continuously differentiable function and
lim f(x)+f'(x)=L
x--> inf
prove
lim f(x)=L
x-->inf

Answer 1

I can think of an example where this is true.
Let \(\bf f(x) = e^{-x}\) , then \(\bf f'(x) = -e^{-x}\)

Answer 2

$$
\Large{ \lim_ {x \to \infty} e^{-x} + (-e^{-x}) = 0 - 0 = 0


}

$$

Answer 3

we can make it more interesting

Answer 4

$$ \Large \rm{
Suppose
\\ f(x) = e^{-x} + k \\
f ' (x) = -e^{-x} \\
}
\Large{
Then \\
\lim_ {x \to \infty} ~ f(x) + f ' (x) \\
=\lim_ {x \to \infty} (e^{-x}+k) + (-e^{-x})
\\ = (e^{-\infty }+k)+ (- e^{-\infty } )
\\= (\frac{1}{e^{\infty }}+ k) + (-\frac{1}{e^{\infty }})
\\~\\
= (0 + k ) + (-0)
= k


}

$$

Answer 5

thats not a proof, that was a plausibility check for the statement. it looks like its true.

Answer 6

your statement is equivalent to proving that
lim(x->oo) f ' (x) = 0

Answer 7

the only way for lim f ' (x) to converge as x-> infinity, is for the limit to go to zero

Answer 8

otherwise the limit will oscillate or diverge