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Which of the following functions are continuous at 
x=-1 ?

g(x)=sin(x+1)

f(x)=(1)/(x-1)
Choose 1 answer:
(A) 
g only
(B) 
f only
(C) Both 
g and 
f
(D) Neither 
g nor 
f

Which of the following functions are continuous at x=1 x=-1 ?\newlineg(x)=sin(x+1) g(x)=\sin (x+1) \newlinef(x)=1x1 f(x)=\frac{1}{x-1} \newlineChoose 11 answer:\newline(A) g g only\newline(B) f f only\newline(C) Both g g and f f \newline(D) Neither g g nor f f

Full solution

Q. Which of the following functions are continuous at x=1 x=-1 ?\newlineg(x)=sin(x+1) g(x)=\sin (x+1) \newlinef(x)=1x1 f(x)=\frac{1}{x-1} \newlineChoose 11 answer:\newline(A) g g only\newline(B) f f only\newline(C) Both g g and f f \newline(D) Neither g g nor f f
  1. Check Function Continuity: To determine if g(x)=sin(x+1)g(x) = \sin(x+1) is continuous at x=1x = -1, we need to check if the function is defined at that point and if the limit as xx approaches 1-1 exists and is equal to the function's value at x=1x = -1.
  2. Calculate g(1)g(-1): Substitute x=1x = -1 into g(x)g(x) to find g(1)g(-1).
    g(1)=sin((1)+1)=sin(0)=0.g(-1) = \sin((-1) + 1) = \sin(0) = 0.
    Since the sine function is continuous everywhere, the limit as xx approaches 1-1 of g(x)g(x) is also 00.
  3. Check Function Continuity: To determine if f(x)=1(x1)f(x) = \frac{1}{(x-1)} is continuous at x=1x = -1, we need to check if the function is defined at that point and if the limit as xx approaches 1-1 exists and is equal to the function's value at x=1x = -1.
  4. Calculate f(1)f(-1): Substitute x=1x = -1 into f(x)f(x) to find f(1)f(-1).
    f(1)=1((1)1)=12=12f(-1) = \frac{1}{((-1) - 1)} = \frac{1}{-2} = -\frac{1}{2}.
    Since the function is defined at x=1x = -1 and the denominator is not zero, the function is continuous at that point.
  5. Verify Continuity: Since both g(x)g(x) and f(x)f(x) are continuous at x=1x = -1, the correct answer is (C) Both gg and ff.

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