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]?

Question

Charles Anctil · Accepted Answer

There is no one sentence answer.

In order for #g# to be continuous on #[3,5]#, the definition of continuous on a closed interval requires: For #c# in #(3,5)#, we need#lim_(xrarrc) g(x) = g(c)# and we also need one-sided continuity at the endpoints: we need: # lim_(xrarr3^+) g(x) = g(3)# and #lim_(xrarr5^-) g(x) = g(5)# For #c# in #(3,5)#, We'll use the properties of limits to evaluate the limit: #lim_(xrarrc) g(x) = lim_(xrarrc) sqrt(-x^2+8x-15)# #= 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 #c=3#, replace all limits of the form #lim_(xrarrc)# with #lim_(xrarr3^+)# For #c=5#, replace all limits of the form #lim_(xrarrc)# with #lim_(xrarr5^-)#

Aurora Alvarado · Answer

undefined To show that the function g(x) = sqrt(-x^2 + 8*x - 15) is continuous on the interval [3,5], we need to demonstrate that it satisfies the definition of continuity and the properties of limits.

First, let's consider the definition of continuity. A function is continuous at a point if three conditions are met: the function is defined at that point, the limit of the function as x approaches that point exists, and the limit is equal to the value of the function at that point.

Next, we need to examine the properties of limits. One important property is that the limit of a sum or difference of functions is equal to the sum or difference of their limits, provided the individual limits exist. Another property is that the limit of a product of functions is equal to the product of their limits, again assuming the individual limits exist. Additionally, the limit of a composition of functions is equal to the composition of their limits, given that the individual limits exist.

Now, let's apply these concepts to the function g(x) = sqrt(-x^2 + 8*x - 15) on the interval [3,5].

First, we need to check if the function is defined on the interval [3,5]. Since the function is a square root, it is defined as long as the expression inside the square root is non-negative. Evaluating the expression -x^2 + 8*x - 15 on the interval [3,5], we find that it is non-negative, so the function is defined on [3,5].

Next, we need to show that the limit of g(x) as x approaches any point within the interval [3,5] exists and is equal to the value of g(x) at that point. This can be done by evaluating the limit algebraically or using properties of limits.

By applying the properties of limits and algebraic manipulations, we can simplify the expression of g(x) to g(x) = sqrt(-(x-3)(x-5)). This form allows us to see that the function is a composition of the square root function and the quadratic function -(x-3)(x-5).

Since the square root function is continuous for non-negative values, and the quadratic function -(x-3)(x-5) is also continuous, we can conclude that the composition of these functions, g(x), is continuous on the interval [3,5].

Therefore, using the definition of continuity and the properties of limits, we have shown that the function g(x) = sqrt(-x^2 + 8*x - 15) is continuous on the interval [3,5].