Find the discontinuities of the given function. Classify them as a point discontinuity or a jump discontunuity.
g(x)= x, x<-3
5x/(x^2-4x), x>= -3
Evaluate g(x) at points close to the discontinuity:
Lim x-> -3- g(x) = -3
g(-3)=5(-3)/((-3)^2-4(-3))
=-15/21
=-5/7
So there is a jump from -3 on the left to -5/7 on the right.
Can you suggest what kind of discontinuity it is?
I assume it's a jump since it jumps from one value to another.
That is correct.
If you do not have clear definitions of each type of discontinuity, you can check the following link:
http://en.wikipedia.org/wiki/Classification_of_discontinuities
To find the discontinuities of the given function, you need to look for values of x where the function is not defined or where there is a sudden jump in the graph.
First, let's look at the first part of the function, g(x) = x for x < -3. This is a line with a slope of 1 passing through the point (-3, -3). There are no discontinuities in this interval.
Now, let's examine the second part of the function, g(x) = 5x/(x^2 - 4x) for x ≥ -3. To find the discontinuities, we need to look for values of x that make the denominator zero or undefined.
Setting the denominator equal to zero, we have (x^2 - 4x) = 0. Factoring out an x, we get x(x - 4) = 0. Solving for x, we find x = 0 and x = 4.
So, x = 0 and x = 4 are the values that make the denominator zero and therefore cause the function to be undefined.
To classify the discontinuities, we examine the limit of the function as x approaches the values that make the function undefined.
As x approaches 0 from the right side (x → 0+), the function approaches positive infinity. As x approaches 0 from the left side (x → 0-), the function approaches negative infinity. This is considered a jump discontinuity.
As x approaches 4 from the right side (x → 4+), the function approaches positive infinity. As x approaches 4 from the left side (x → 4-), the function approaches negative infinity. This is also considered a jump discontinuity.
Therefore, the function g(x) = x for x < -3 has no discontinuities, and the function g(x) = 5x/(x^2 - 4x) for x ≥ -3 has jump discontinuities at x = 0 and x = 4.