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Determine if continuous with work.

Determine if continuous with work.-example-1
User Gongqj
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Answer and Explanation:

We can determine if the piecewise function:


g(x)=\begin{cases}(x^2-a^2)/(x-a) &\text{ for }x\\e a \\ 8 &\text{ for } x = a\end{cases}

is continuous for all values of x by looking at the following conditions:


  1. g(a) is defined

  2. \lim_(x\to a)g(x) = g(a)

To evaluate the first condition, we can look to which piece of the function operates on the domain including
x=a. We can see that this piece is:


g(x) = 8\text{ for } x=a

Therefore, the first condition is satisfied.

To evaluate the second condition, we can take take the limit of the other piece of the function as x approaches a, and make sure that it approaches the actual value of
g(x) at x = a, which is 8.


\lim_(x\to a)\!\left((x^2-a^2)/(x-a)\right)

factoring the numerator within the limit


\lim_(x\to a)\!\left(((x+a)(x-a))/(x-a)\right)

canceling the common term in the numerator and denominator (x - a) to fill in the removable discontinuity


\lim_(x\to a)(x+a)

evaluating the limit


a + a

executing the addition


\lim_(x\to a)g(x) = 2a

Now, we can compare this to the actual value of
g(x) at x = a:


2a \stackrel{?}= 8

dividing both sides by 2


a\stackrel{?}=4

Finally, we can see that the function is only continuous for all values of x if
a = 4.

User Joon
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