Math

QuestionIs the piecewise function f(x)={2xif x22xif x<2f(x)=\left\{\begin{array}{ll} 2|x| & \text{if } x \geqslant -2 \\ 2x & \text{if } x < -2 \end{array}\right. continuous at x=cx=c, where c=2c=-2?

Studdy Solution

STEP 1

Assumptions1. The function is defined as a piecewise function, with two different definitions for xx \geqslant - and x<x < -. . We are checking for continuity at x=cx=c, where c=c=-.

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) exists4. limxcf(x)=f(c)=limxc+f(x)\lim_{x\to c^-}f(x) = f(c) = \lim_{x\to c^+}f(x)

STEP 3

First, we need to find f(c)f(c), where c=2c=-2. Since c=2c=-2 falls in the range x2x \geqslant -2, we use the first definition of the piecewise function to find f(c)f(c).
f(c)=2cf(c) =2|c|

STEP 4

Plug in the value of c=2c=-2 to find f(c)f(c).
f(c)=22f(c) =2|-2|

STEP 5

Calculate f(c)f(c).
f(c)=22=4f(c) =2|-2| =4

STEP 6

Next, we need to find limxcf(x)\lim_{x\to c^-}f(x). Since xx approaches c=2c=-2 from the left, we use the second definition of the piecewise function to find this limit.
limxcf(x)=2x\lim_{x\to c^-}f(x) =2x

STEP 7

Plug in the value of c=2c=-2 to find limxcf(x)\lim_{x\to c^-}f(x).
limxcf(x)=2(2)\lim_{x\to c^-}f(x) =2(-2)

STEP 8

Calculate limxcf(x)\lim_{x\to c^-}f(x).
limxcf(x)=2(2)=4\lim_{x\to c^-}f(x) =2(-2) = -4

STEP 9

Now, we need to find limxc+f(x)\lim_{x\to c^+}f(x). Since xx approaches c=2c=-2 from the right, we use the first definition of the piecewise function to find this limit.
limxc+f(x)=2x\lim_{x\to c^+}f(x) =2|x|

STEP 10

Plug in the value of c=2c=-2 to find limxc+f(x)\lim_{x\to c^+}f(x).
limxc+f(x)=22\lim_{x\to c^+}f(x) =2|-2|

STEP 11

Calculate limxc+f(x)\lim_{x\to c^+}f(x).
limxc+f(x)==4\lim_{x\to c^+}f(x) =|-| =4

STEP 12

Now that we have f(c)f(c), limxcf(x)\lim_{x\to c^-}f(x), and limxc+f(x)\lim_{x\to c^+}f(x), we can check if the function is continuous at x=cx=c. A function is continuous at x=cx=c if limxcf(x)=f(c)=limxc+f(x)\lim_{x\to c^-}f(x) = f(c) = \lim_{x\to c^+}f(x).

STEP 13

Check if limxcf(x)=f(c)=limxc+f(x)\lim_{x\to c^-}f(x) = f(c) = \lim_{x\to c^+}f(x).
==- = =Since limxcf(x)f(c)\lim_{x\to c^-}f(x) \neq f(c), the function is not continuous at x=cx=c.
The piecewise function is not continuous at x=cx=c, where c=2c=-2.

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