Where are these examples discontinuous?
Please explain step by step. Thank you.
Please explain step by step. Thank you.
1 Answer
a. Removable discontinuity at
Explanation:
Discontinuities tend to happen at three distinct points:
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At vertical asymptotes (infinite discontinuities), which can be found by setting the denominator of the rational function equal to zero and solving for
#x.# -
At holes (removable discontinuities), which come to exist by canceling out a common factor in both the numerator and dominator.
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At jumps, which generally happen in the graphs of piecewise functions, usually resulting in a notable gap between two curves in the graph.
Keeping these in mind, let's proceed:
a. Let's simplify a little. The denominator can be factored, yielding:
Setting the denominator equal to zero and solving yields
So, there's an infinite discontinuity (vertical asymptote) at
b. Again, the first thing we do is simplify. There's not much that can be done with the numerator; however, we factor the denominator (it's a quadratic), yielding
This tells us there is a removable continuity at
The simplified form is
c. This function is much unlike the other two. The first objective should be to get rid of the absolute value and get a piecewise function instead. We do this by realizing that
So, we really have a piecewise function in the form
It appears that we have removable discontinuities, but upon closer inspection, this is not the case. Yes,