Question

Show by means of an example that lim x-> a [f(x) + g(x)] may exist even though neither lim x-> a f(x) and lim x -> a g(x) exists.

Answer 1

hey

Answer 2

The easiest example I can think of is a fraction that equals a constant except at a particular place. And then separate the fraction.

Answer 3

Example give me a fraction that doesn't exist at x=1 but exist everywhere else that is a constant value.

Answer 4

You can choose that constant value to be 1 if you want...

Answer 5

choose a fraction that equals 1 everywhere except at one value of x...

Answer 6

oh maybe you aren't here

Answer 7

\(\color{#000000 }{ \displaystyle \lim _{x\to {\tiny}3}\left[\frac{x}{x-3}+\frac{-3}{x-3}\right] }\)

Notice that both of the limits diverge (to \(\small \pm\infty\))
if you take these limits individually/separately.

HOWEVER, THE LIMIT OF THE SUM EXISTS!\(\tiny \\[0.8em]\)
\(\color{#000000 }{ \displaystyle \lim _{x\to {\tiny}3}\left[\frac{x}{x-3}+\frac{-3}{x-3}\right] =\lim _{x\to {\tiny}3}\left[\frac{x-3}{x-3}\right] =1}\)