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QuestionFind lim⁡x→3f(x)\lim _{x \rightarrow 3} f(x) for the piecewise function f(x)={x2−7x>3−4+2xx<3f(x)=\begin{cases} x^{2}-7 & x>3 \\ -4+2x & x<3 \end{cases}.

Studdy Solution

STEP 1

Assumptions1. The function f(x)f(x) is defined as a piecewise function - f(x)=x−7f(x) = x^{}-7 for x>3x>3 - f(x)=−4+xf(x) = -4+x for x<3x<3 . We are asked to find lim⁡x→3f(x)\lim{x \rightarrow3} f(x).

STEP 2

The limit of a function as xx approaches a certain value can be different from the left and right. To find the limit at x=x =, we need to calculate both the left-hand limit and the right-hand limit.

STEP 3

First, let's calculate the left-hand limit, which is the limit of f(x)f(x) as xx approaches 33 from the left. This is denoted as lim⁡x→3−f(x)\lim{x \rightarrow3^-} f(x).

STEP 4

For x<3x<3, f(x)=−4+2xf(x) = -4+2x. So, we substitute 33 into this equation to find the left-hand limit.
lim⁡x→3−f(x)=−4+2(3)\lim{x \rightarrow3^-} f(x) = -4+2(3)

STEP 5

Calculate the left-hand limit.
lim⁡x→3−f(x)=−4+2(3)=2\lim{x \rightarrow3^-} f(x) = -4+2(3) =2

STEP 6

Next, let's calculate the right-hand limit, which is the limit of f(x)f(x) as xx approaches 33 from the right. This is denoted as lim⁡x→3+f(x)\lim{x \rightarrow3^+} f(x).

STEP 7

For x>3x>3, f(x)=x2−7f(x) = x^{2}-7. So, we substitute 33 into this equation to find the right-hand limit.
lim⁡x→3+f(x)=(3)2−7\lim{x \rightarrow3^+} f(x) = (3)^{2}-7

STEP 8

Calculate the right-hand limit.
lim⁡x→3+f(x)=(3)2−7=2\lim{x \rightarrow3^+} f(x) = (3)^{2}-7 =2

STEP 9

Now, we compare the left-hand limit and the right-hand limit. If they are equal, then the limit of the function at x=3x =3 exists and is equal to this common value. If they are not equal, then the limit does not exist (NE).

STEP 10

Compare the left-hand limit and the right-hand limit.
lim⁡x→3−f(x)≠lim⁡x→3+f(x)\lim{x \rightarrow3^-} f(x) \neq \lim{x \rightarrow3^+} f(x)Since the left-hand limit and the right-hand limit are not equal, the limit of the function at x=3x =3 does not exist.
So, lim⁡x→3f(x)\lim{x \rightarrow3} f(x) = DNE

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