Question

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

Answer 1

E

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) (f(x))/(g(x)+6)`

This is the same as:

`( \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:

`( \lim_(x \to 5) f(x))/( \lim_(x \to 5) g(x)+6)`

Substitute the values we know:

`=((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) (f(x)-2)/(g(x))`

Again, this is the same as:

`( (\lim_(x \to 5) f(x))-2)/( \lim_(x \to 5) g(x))`

Substitute in the values we know:

`=((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.