Question

HELP!!!!!!!!!! AP CALCULUS AB

Answer 1

Answer:

$\lim_{x \to 5^{-}} f(x)= 54$

$\lim_{x \to 5^{+}} f(x) = -17$

Step-by-step explanation:

+ means we are approaching from the right

- means we are approaching from the left

We are given that if x < 5, it is x² + 5x + 4

So to find $\lim_{x \to 5^{-}} f(x)$ we would plug in 5 into that piecewise function part:

5² + 5(5) + 4 = 54

We would be approaching y = 54 if we approach from the left.

We are given that x > 5, it is -4x + 3

So to find $\lim_{x \to 5^{+}} f(x)$ we would plug in 5 into that piecewise function part:

-4(5) + 3 = -17

We would approach y = -17 if we approached from the right.

Answer 2

Answer:

$\lim_{x \to 5^-}f(x)=54\\\lim_{x \to 5^+}f(x)=-17$

Step-by-step explanation:

So, we have the function:

$f(x)=x^2+5x+4,x$

And we want to show that this has a jump discontinuity.

As directed, calculate the limit at x=5 from the left and from the right. Thus:

$\lim_{x \to 5^-}f(x)$

To calculate this, since we're coming from the left, x is less than 5. So, use the first equation:

$\lim_{x \to 5^-}f(x)\\= \lim_{x \to 5^-}(x^2+5x+4)$

Use direct substitution:

$=(5)^2+5(5)+4\\=25+25+4\\=54$

Thus, as the limit approaches f(5) from the left, the function approaches 54.

Now, calculate the limit at x=5 from the right. Since we're coming from the right, use the third equation:

$\lim_{x \to 5^+}f(x)\\= \lim_{x \to 5^+}(-4x+3)$

Direct substitution:

$(-4(5)+3)\\=-20+3\\=-17$

Thus, as the limit approaches f(5) from the right, the function approaches -17.

As you can picture, there's a huge gap between y=54 and y=-17. The limits are not equal and both of the limits do exist. Thus, we have jump discontinuity.