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Triss [41]
3 years ago
11

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

Mathematics
2 answers:
raketka [301]3 years ago
8 0

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.

Shtirlitz [24]3 years ago
5 0

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 <em>right</em>, 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.

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