Math

QuestionIs the piecewise function f(x)={x2if x33if x<3f(x)=\begin{cases} \sqrt{x^{2}} & \text{if } x \geqslant-3 \\ -3 & \text{if } x<-3 \end{cases} continuous at x=3x=-3?

Studdy Solution

STEP 1

Assumptions1. The function is defined as f(x)={xif x33if x<3f(x)=\begin{cases}\sqrt{x^{}} & \text{if } x \geqslant-3 \\-3 & \text{if } x<-3\end{cases} . We are asked to determine if the function is continuous at x=c=3x=c=-3

STEP 2

A function is continuous at a point x=cx=c if the following three conditions are met1. f(c)f(c) is defined2. limxcf(x)\lim{{x \to c^-}} f(x) exists. limxc+f(x)\lim{{x \to c^+}} f(x) exists and limxcf(x)=limxc+f(x)=f(c)\lim{{x \to c^-}} f(x) = \lim{{x \to c^+}} f(x) = f(c)We will check these conditions for x=c=x=c=-.

STEP 3

First, we check if f(c)f(c) is defined at x=c=3x=c=-3. From the definition of the function, we see that f(3)f(-3) is defined and equals (3)2\sqrt{(-3)^{2}}.
f(3)=(3)2f(-3) = \sqrt{(-3)^{2}}

STEP 4

Calculate the value of f(3)f(-3).
f(3)=(3)2=3f(-3) = \sqrt{(-3)^{2}} =3

STEP 5

Now, we check if the left-hand limit, limx3f(x)\lim{{x \to -3^-}} f(x), exists. For x<3x<-3, the function is defined as f(x)=3f(x)=-3.
limx3f(x)=limx33\lim{{x \to -3^-}} f(x) = \lim{{x \to -3^-}} -3

STEP 6

Calculate the left-hand limit.
limx33=3\lim{{x \to -3^-}} -3 = -3

STEP 7

Next, we check if the right-hand limit, limx3+f(x)\lim{{x \to -3^+}} f(x), exists. For x3x \geqslant -3, the function is defined as f(x)=x2f(x)=\sqrt{x^{2}}.
limx3+f(x)=limx3+x2\lim{{x \to -3^+}} f(x) = \lim{{x \to -3^+}} \sqrt{x^{2}}

STEP 8

Calculate the right-hand limit.
limx3+x2=(3)2=3\lim{{x \to -3^+}} \sqrt{x^{2}} = \sqrt{(-3)^{2}} =3

STEP 9

Now that we have the values of f(3)f(-3), limx3f(x)\lim{{x \to -3^-}} f(x), and limx3+f(x)\lim{{x \to -3^+}} f(x), we can compare them to check the continuity of the function at x=3x=-3.
From our calculations, we havef(3)=3f(-3) =3
limx3f(x)=3\lim{{x \to -3^-}} f(x) = -3
limx3+f(x)=3\lim{{x \to -3^+}} f(x) =3
Since limx3f(x)f(3)\lim{{x \to -3^-}} f(x) \neq f(-3), the function is not continuous at x=3x=-3.

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