# How do you use the definition of continuity and the properties of limits to show that the function #g(x) = sqrt(-x^2 + 8*x - 15)# is continuous on the interval [3,5]?

##### 1 Answer

There is no one sentence answer.

#### Explanation:

In order for

For

and we also need one-sided continuity at the endpoints:

we need:

For

#= sqrt(lim_(xrarrc)(-x^2+8x-15))#

#= sqrt(lim_(xrarrc)(-x^2)+lim_(xrarrc)(8x)-lim_(xrarrc)(15))#

#= sqrt(-lim_(xrarrc)(x^2)+8lim_(xrarrc)(x)-lim_(xrarrc)(15))#

#= sqrt(-(lim_(xrarrc)(x))^2+8lim_(xrarrc)(x)-lim_(xrarrc)(15))#

#= sqrt(-(c)^2+8(c)-(15))#

#= g(c)#

Use the one-sided versions of the limit properties at the endpoints.

For

For