Determine the value(s) of x x x (if any) for which the function is discontinuous.

this problem we want to determine the values of x, if any, for which this function is discontinuous. Now we're given a piecewise function here, which means that we wanna take a look at the x values in between each of our pieces. Now here we only have 2 pieces, so that means we only have one x value to worry about, and that x value is x equals negative 1, the transitional point between the two pieces of my piecewise function. Now remember that whenever we're working with discontinue or trying to find discontinuities, we wanna check if our limit is equal to our function value. Because if they are not equal to each other, we know that we are facing a discontinuity. So for our value x equals negative one, let's go ahead and start by finding our limit. So we wanna look at the limit of our function f of x as x approaches negative 1. Now because this is a piecewise function, we wanna make sure that we look at our one-sided limits here in order to determine our regular limit. So let's first look at our left sided limit. So the limit of our function f of x as x approaches negative one from the left. Now from the left side, looking at my function here, my function is gonna be equal to this quadratic, 3x squared plus 4x plus 5. So I wanna go ahead and plug negative one into my function here. So I have 3 times negative one squared, plus 4 times negative one, plus 5. Now doing this algebra here, 3 times negative one squared is just going to be 3. 4 times negative one is negative 4, and then a plus 5. So 3 minus 4 plus 5 is going to end up being equal to positive 4. So that left sided limit is 4. Now let's take a look at our right sided limit. So the limits of my function f of x as x approaches negative one from that right side. Now looking at my piecewise function, I know that coming in from the right side, my function is just going to be equal to 4. So I already know that that right sided limit is just 4. Now since my left side of limit and my right sided limit are both 4, then I know that my regular limit, my limit of f of x as x approaches negative one is also 4. So now that we have that limit, let's check our function value. So our function value f of negative one. Looking up at my functions here, I know that for x greater than or equal to negative one, my function is just equal to 4. So that tells me that at negative one, my function value is 4. And here we see that our limit is the exact same as our function value. These 2 are equal. So that tells me that my function is actually continuous here. Now because it's continuous at the only point that I was sort of worried about in between these two pieces, that means that my function is actually continuous everywhere and I don't have any discontinuities. So since this question asked me to determine the values, if any, for which my function is discontinuous, I don't have any values of x for which it's discontinuous because this function is fully continuous everywhere. Now this might be surprising to see a piecewise function that doesn't have a discontinuity, but this makes total sense. Because sometimes when we look at piecewise functions, there is no jump. There is no hole. Sometimes they continue on seamlessly. Just now they look like a different function. They have a different shape. So it's possible for a piecewise function to be continuous everywhere. Let me know if you have any questions here. Thanks for watching.

Determine the value(s) of x x x (if any) for which the function is discontinuous.

x = 4 , x = 5 x=4,x=5 x = 4 , x = 5

x = 5 , x = 1 x=5,x=1 x = 5 , x = 1

1. Limits and Continuity

Continuity

Business Calculus

1. Limits and Continuity

Continuity

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Multiple Choice

Determine the value(s) of $x$ (if any) for which the function is discontinuous.

$x=4,x=5$

$x=5,x=1$

$x=−1$

Function is continuous everywhere

Verified step by step guidance

Step 1: Understand the definition of continuity. A function is continuous at a point if the following three conditions are satisfied: (1) The function is defined at the point, (2) The limit of the function exists at the point, and (3) The value of the function at the point equals the limit of the function at the point.

Step 2: Analyze the given piecewise function. The function is defined as f(x) = 3x^2 + 4x + 5 for x < -1 and f(x) = 4 for x ≥ -1. The potential point of discontinuity is at x = -1, where the definition of the function changes.

Step 3: Check the left-hand limit of the function as x approaches -1 from the left (x < -1). Substitute x = -1 into the expression 3x^2 + 4x + 5 to find the value of the left-hand limit.

Step 4: Check the right-hand limit of the function as x approaches -1 from the right (x ≥ -1). Since the function is defined as f(x) = 4 for x ≥ -1, the right-hand limit is simply 4.

Step 5: Compare the left-hand limit, right-hand limit, and the value of the function at x = -1. If all three are equal, the function is continuous at x = -1. If they are not equal, the function is discontinuous at x = -1.