Question

Given the piecewise function above evaluate the following statements. (write "DNE" if the limit does not exist)

Answer 1

By analyzing the graph of the function, we will see that:

• a) DNE
• b) 3
• c) -1
• d) 3
• e) DNE
• f) 2

How to get the limits of the given function?

Remember that when we take a limit to a given value, we approach to that value by two directions (from above and below) and see at which value the function approaches.

So, for the first limit we have:

x ⇒ -2

As you can see, if we approach from below the function tends to 4, while if we approach from above, the function tends to -1, which means that this limit does not exist. DNE.

x ⇒ 0

The function is continuous here, so the limit is equal to the value of the function in that point, which is 3

x ⇒ 2

If we approach from above and below we get the same number, which is -1, so the limit is equal to -1.

x = 2

Now we are directly evaluating the function in x = 2, and we can see that there is a jump such that f(2) = 3

x ⇒ 4

Here we have two asymptotes, one goes to infinity and the other to negative infinity, so this limit does not exist.

x ⇒ ∞

So we can't see how the function behaves for large values of x, so there is no way to answer this exactly.

If we suppose that the function does not change, we can see that we have a rational function in the right side of the image, so we can expect that as x tends to infinity, the function will tend asymptotically to 2.

Why 2?

Because the denominator becomes zero when x = 4, then when x = 5 the denominator is equal to 1.

At that point, the function is equal to 3, so the function is something like:

y = 1/(x - 4) + 2

If you want to learn more about limits, you can read:

brainly.com/question/5313449