use the definition of continuity to determine whether f(x)=(x^2 + 7)/(x - 4) is continuous at 5. is f…

use the definition of continuity to determine whether f(x)=(x^2 + 7)/(x - 4) is continuous at 5. is f continuous at 5? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. a. no. because f(5) does not exist. b. yes. because f(5)= and lim f(x)= as x→5. c. no. because f(5)= and lim f(x)= as x→5, and f(5)≠lim f(x) as x→5.

use the definition of continuity to determine whether f(x)=(x^2 + 7)/(x - 4) is continuous at 5. is f continuous at 5? select the correct choice below and, if necessary, fill in the answer boxes to complete your answer choice. a. no. because f(5) does not exist. b. yes. because f(5)= and lim f(x)= as x→5. c. no. because f(5)= and lim f(x)= as x→5, and f(5)≠lim f(x) as x→5.

Answer

Explanation:

Step1: Calculate f(5)

Substitute (x = 5) into (f(x)=\frac{x^{2}+7}{x - 4}). [f(5)=\frac{5^{2}+7}{5 - 4}=\frac{25 + 7}{1}=32]

Step2: Calculate (\lim_{x\rightarrow5}f(x))

Since (f(x)=\frac{x^{2}+7}{x - 4}) is a rational - function and the denominator is non - zero at (x = 5), we can directly substitute (x = 5) into the function. (\lim_{x\rightarrow5}f(x)=\frac{5^{2}+7}{5 - 4}=32)

Step3: Check the continuity condition

A function (y = f(x)) is continuous at (x=a) if (\lim_{x\rightarrow a}f(x)=f(a)). Here (a = 5), (f(5)=32) and (\lim_{x\rightarrow5}f(x)=32).

Answer:

B. Yes. Because (f(5)=32) and (\lim_{x\rightarrow5}f(x)=32)