Question

Use the table for each problem to find the given limits.

Answer 1

1)
`\lim_(x \to3 ) (2f(x))+g(-x))=13`

2)
`\lim_(x \to3 )(g(x))/(f(-x))=1/2`

Explanation:

So we are given the limits:

`\lim_(x \to3 )f(x)=4\text{ and } \lim_(x \to-3 ) f(x)=2`

And:

`\lim_(x \to 3 ) g(x)= 1\text{ and } \lim_(x \to -3 ) g(x)=5`

Question A)

We have the limit:

`\lim_(x \to3 ) (2f(x))+g(-x))`

We can split this limit using our properties:

`= \lim_(x \to 3) (2f(x))+\lim_(x \to 3) g(-x)`

Now, use direct substitution. Substitute 3 for x. So:

`=2(f(3))+g(-3)`

We are given that f(3) (or the limit as x approaches towards 3) is 4.

We know that the limit as x tends towards -3 of g(x) is 5. In other words, g(-3) can be said to be 5. So:

`=2(4)+(5)`

Multiply:

`=8+5=13`

So, our limit is:

`\lim_(x \to3 ) (2f(x))+g(-x))=13`

Question B:

We have the limit:

`\lim_(x \to3 )(g(x))/(f(-x))`

Again, we can rewrite this as:

`(\lim_(x \to3 )g(x))/(\lim_(x \to3 )f(-x))}`

Direct substitution:

`=(g(3))/(f(-3))`

The value in the numerator, as given, is 1.

The value in the denominator will be 2. So:

`=1/2`

Therefore, our limit is:

`\lim_(x \to3 )(g(x))/(f(-x))=1/2`

And we're done!