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Problem 15 Medium Difficulty

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

$ f(x) = x + \sqrt{x - 4}, \hspace{5mm} [4, \infty) $

Answer

So $f$ is continuous at $x=a$ for every $a$ in $(4, \infty) .$ Also, $\lim _{x \rightarrow 4^{+}} f(x)=4=f(4),$ so $f$ is continuous from the right at 4.
Thus, $f$ is continuous on $[4, \infty)$

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Video Transcript

Yeah, okay. Uh We are looking at a function F of X equals X plus the square root of x minus four on the interval. Uh For closed, half closed half open interval, uh X can equal for all the way up towards infinity. Now for f of X to be continuous at a point half of s F of x is continuous. F of X will be continuous at a point X equals A. If the limit of our function F a bex as X approaches the number A is equal to the value of our function mm X equals A. So if the limit of F of X as X approaches A is equal to f of a, then at this continuous at X equals a. Uh This is going to be a continuous function on uh this interval for to infinity. The only time you could possibly run into trouble with dysfunction is if you were trying to take the square root of a negative number, which of course you cannot do but you're taking the square root of X -4. And uh if X equals for member X is restricted to this interval, if x equals four, then four minus 40 and the square with zero is zero. So that's okay. And for any other value of X on this interval, X is going to be greater than four. And so x minus four will be a positive number and you can always take the square root of a positive number. So this is going to be a continuous function on this interval. And so the limit of this function as X approaches any number a belonging to this interval will equal the value of this function at that point. A um I also have this function graft, uh so that you can see that it is continuous on this interval for to infinity. So here is the graph of the function x plus the square root of X -4. And over here, this is the point when X is four, I don't know if it will let me plot that point there. It is right there. Yeah. All right. So when X equals four, um the function is defined and for any value of X on the interval from four towards positive infinity, you can see that dysfunction is continuous and that as X approaches any particular number, uh to function will be approaching F of a or F of that number.