Question

If lim f (x) x →5 = 2 and lim g (x) x → 5= -6 which of these limits exist?

Answer 1

Answer:

E

Step-by-step explanation:

So we already know that:

$\lim_{x \to 5} f(x)=2 \text{ and } \lim_{x \to 5} g(x)=-6$

So, go through each of the choices and see which ones are correct.

I)

We have:

$\lim_{x \to 5} (f(x)+g(x))$

This is the same as saying:

$\lim_{x \to 5} f(x)+ \lim_{x \to 5} g(x)$

And since we already know the values:

$=(2)+(-6)\\=-4$

So, the limit does indeed exist.

II)

We have:

$\lim_{x \to 5} \frac{f(x)}{g(x)+6}$

This is the same as:

$\frac{ \lim_{x \to 5} f(x)}{ \lim_{x \to 5} g(x)+ \lim_{x \to 5} 6 }$

The bottom right one is just 6. Simplify:

$\frac{ \lim_{x \to 5} f(x)}{ \lim_{x \to 5} g(x)+6}$

Substitute the values we know:

$=\frac{(2)}{(-6)+6} \\=2/0$

This is a value over zero. Unlike the indeterminate form 0/0, this limit does not exist.

III)

We have:

$\lim_{x \to 5} \frac{f(x)-2}{g(x)}$

Again, this is the same as:

$\frac{ (\lim_{x \to 5} f(x))-2}{ \lim_{x \to 5} g(x)}$

Substitute in the values we know:

$=\frac{(2)-2}{(-6)}\\ =0/-6=0$

The limit does exist and it is 0.

So, the limits of only I and III exist.

The correct answer is E.