Find the intervals on which the following functions are contin...

Question

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.

$h (x) = \frac{2 x}{x^{3} - 25 x}$

Accepted Answer

Welcome back everyone. In this problem, we want to determine the intervals of continuity for the function G of X equals 4 X divided by X minus 49 X. A says they are sorry, parentheses 70 closed parentheses unioned with parentheses 07 closed parentheses. Be open parenthesis negative infinity 7 closed parentheses union open parentheses 70 closed parentheses union open parentheses 07 closed parentheses union open parentheses 7 infinity closed parentheses. C says its open parentheses negative infinity -7 closed parentheses union open parentheses 7 infinity closed parenthesis and D says its open parentheses negative infinity 7 closed bracket union open bracket 7 infinity closed parentheses. Now how are we going to determine the intervals of continuity? Well, let's think in the reverse. At what points are our function discontinuous. If we can figure out what those are, then we'll know on what intervals it will be continuous. So let's start by trying to figure that out. Notice that geoff X is a quotient, so we know that it's going to be discontinuous where the denominator is 0 because division by zero is undefined. OK? So, Where the denominators. The denominator of geoX. Equals 0, OK. The function is undefined. And no, that basically tells us then that we are trying to solve for the values of X, for which X cubed minus 49 X is going to be equal to 0. Now, let's start by first trying to express this as a linear expression, OK? And no, notice we can factor X from both terms, OK. So that's going to be X multiplied by X2 minus 49 equals 0. This expression is the difference of 2 squares, so we can even take it a step further to rewrite this as X multiplied by X + 7 multiplied by X minus 7 being equal to 0. And no, this tells us then that X is going to be equal to 0, X + 7 is going to be equal to 0, and X minus 7 equals 0. Already X equals 0 is a solution. If X + 7 equals 0, that means X will be equal to -7. And if x minus 7 equals 0, that means X equals 7. So this tells us that X equals 0, X equals -7, and X equals 7 are the points of discontinuity. Now that we have the points of discontinuity, how can we determine the intervals of continuity? Well, let's just do a quick sketch of the graph to help us understand what exactly we're looking for here. So, uh, let me draw our X and Y axis, OK. And no, really, this graph is probably going to look something like this, OK. Something like that. Where we know that it's discontinuous at. -7, OK, 0 and 7. So since it's discontinuous at that point here, we noticed that we have our vertical asymptote uh for our term here, that means there will be an empty space. Where X equals 0, it's discontinuous. Let me, let me actually move the 0 here to make it easier to read, OK. And it's also going to be the same at X equals 7. So now, we are trying to figure out what these intervals will be on our X axis. In other words, what is going to be our interval here while it's continuous right here, here and here, OK? And of course this is going from negative infinity. To positive infinity. So I'll just write infinity here, but we know we're referring to our X axis. And let me write it right here. OK. So that's the question we're trying to figure out. What are these intervals going to be? Well, starting from the left, OK. Notice here that it goes from negative infinity to -7, not including, or not to include -7. So this interval is negative infinity. Let me put that in red. Negative in parenthesis negative infinity -7 closed parentheses. For next interval it goes from -7 to 0, not including both of those, so that's open parentheses -7, 0 closed parentheses. For the next interval. It goes from 0 to 7, which doesn't include the points of discontinuity X equals 0 and X equals 7. So open parentheses 07 close parentheses, and then finally it's going to go from 7 to infinity, not to include X equals 7 as the point of discontinuity. That's parentheses 7 infinity closed parentheses. Therefore, these are going to be our intervals of continuity for the function G of X. If we look back at our answer choices, that means then that B. Is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find the intervals on which the following functions are continuous. Specify right- or left-continuity at the finite endpoints.
$h (x) = \frac{2 x}{x^{3} - 25 x}$

Key Concepts

Continuity of Functions

Identifying Discontinuities

Endpoints and One-Sided Limits

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