Continuity
What is a continuous function?
By definition, it is said that a function is continuous at x = a if the limit as approaches a equals the value of the function at a. The definition of a continuous function is given below:
Definition:
A function is continuous at point a if all off the following are true:
\lim_{x \rightarrow a}f(x) = f(a)
\lim_{x \rightarrow a^+}f(x) = f(a)
\lim_{x \rightarrow a^-}f(x) = f(a)
To further explore the concept of a continuous function, let’s take a look at the rational function shown in the graph.
Let us now try to answer the question, “where would the function NOT be continuous”? Well, we know that rational functions are continuous everywhere except where division by zero occurs.
So, to find where this rational function is not continuous, we need to calculate the location of its vertical asymptotes by setting the denominator equal to zero, as shown below:
x^2-2x-15 = (x-5)(x+3) = 0
This means that the vertical asymptotes are at x = 5 and x = -3. This tells us that the function f(x) = \frac{2x+5}{x^2-2x-15} is not continuous at x = 5 and x = -3.
Example 7
For what values of x is the following function discontinuous?
\Large{\frac{x-1}{2x^2+3x-9}}
Step 1:
Let us first graph the function f(x) = \frac{x-1}{2x^2+3x-9}, as shown below.
Step 2:
Simply by inspecting the graph, we can see that the function has 2 vertical asymptotes. Remember that a rational function is discontinuous at the location of its vertical asymptotes. So, to find the x values where the function is discontinuous, we will find the vertical asymptotes by setting the denominator equal to zero:
\Large{2x^2+3x-9 = 0}
\Large{(2x-3)(x+3) = 0}
Thus, the vertical asymptotes are at x = \frac{3}{2} and x = -3
This means that the function is also discontinuous at x = \frac{3}{2} and x = -3
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