To demonstrate that the function is continuous on the given interval, use the definition of continuity and the characteristics of limits.f(x)=x+x−4,&nbsp;[4,∞)

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To demonstrate that the function is continuous on the given interval, use the definition of continuity and the characteristics of limits.f(x)=x+x−4,&amp;nbsp;[4,∞)

au4gsf · Accepted Answer

If a function is continuous at every point in an interval, the interval is said to be continuous.Now, a function is continuous at a point&amp;nbsp;x=a&amp;nbsp;if&amp;nbsp;limx→af(x)=f(a)The function is&amp;nbsp;f(x)=x+x−4&amp;nbsp;defined on the interval&amp;nbsp;[4,∞).For a&amp;gt;4 we have,limx→af(x)=limx→a(x+x−4)=limx→ax+limx→a(x−4)=limx→ax+limx→a(x−4)limx→ax+limx→a(x)−lima→a(4)=a+a−4=f(a)That is,&amp;nbsp;limx→af(x)=f(a)&amp;nbsp;for all values for a&amp;gt;4. Therefore f is continuous at x=a for every a in&amp;nbsp;(4,∞)Now&amp;nbsp;limx→4+f(x)=limx→4+(x+x−4)=4+4−4=4+0=4=f(4)Also,&amp;nbsp;limx→4+f(x)=4=f(4),&amp;nbsp;so f is continuous the right at 4.Thus, the function&amp;nbsp;f(x)=x+x−4&amp;nbsp;is continuous at all points in the interval&amp;nbsp;[4,∞)Therefore, f is continuous at x=a for every a in&amp;nbsp;(4,∞).Also,limx→4+f(x)=4=f(4), so f is continuous from the right at 4.Thus, the function is continuous on&amp;nbsp;[4,∞).

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Don Sumner · Accepted Answer

Definition of Continuity: A function f(x) is continuous at a point c if the following three conditions are met:1. f(c) is defined.2. limx→cf(x) exists.3. limx→cf(x)=f(c).Using the given function f(x)=x+x−4 on the interval [4,∞), we can verify each condition.1. f(c) is defined:For any x in the interval [4,∞), the expression x+x−4 is defined. Therefore, f(c) is defined for all x in the interval.2. limx→cf(x) exists:To determine the limit as x approaches c, we substitute x=c into the function and simplify:limx→c(x+x−4)=c+c−4.In this case, as x approaches any value greater than or equal to 4, the limit exists.3. limx→cf(x)=f(c):Now we compare the limit as x approaches c to the value of the function at c:limx→c(x+x−4)=c+c−4.f(c)=c+c−4.By comparing these two expressions, we can see that the limit and the value of the function at c are equal.Therefore, since the function f(x)=x+x−4 satisfies all three conditions of continuity, we can conclude that it is continuous on the interval [4,∞).

nick1337 · Accepted Answer

Step 1. f(c) is defined:In our case, f(x)=x+x−4 is defined for all x in the interval [4,∞), including the point x=4. Hence, f(c) is defined.Step 2. The limit of f(x) as x approaches c exists:We need to evaluate the limit of f(x) as x approaches 4 from the right side (x→4+) to check if it exists.limx→4+(x+x−4)To simplify this expression, we can substitute x−4 with t:limt→0+((t+4)+t)Now, we can evaluate the limit:limt→0+(t+4)+limt→0+tThe first term evaluates to 4, and the second term evaluates to 0. Therefore, the limit exists and is equal to 4.Step 3. The limit of f(x) as x approaches c is equal to f(c):We need to check if limx→4+(x+x−4)=f(4).From the previous calculation, we know that limx→4+(x+x−4)=4. Now, let&#039;s find f(4):f(4)=4+4−4=4+0=4+0=4Since limx→4+(x+x−4)=f(4), the third condition is satisfied.Therefore, based on the definition of continuity, we have shown that the function f(x)=x+x−4 is continuous on the interval [4,∞).

Use the definition of continuity and the properties of limits to show that the f

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