Question

Determine if continuous with work.

Answer 1

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`.