Which of the following are true about f?I. lim𝑥→𝑎 𝑓(𝑥) exists.II. 𝑓(𝑎) exists.III.𝑓(𝑥) is continuous at x = a I. x→alim f(x) exists.II. f(a) exists.III.f(x) is continuous at x = a
Question
Which of the following are true about f?I. lim𝑥→𝑎 𝑓(𝑥) exists.II. 𝑓(𝑎) exists.III.𝑓(𝑥) is continuous at x = a I. x→alim f(x) exists.II. f(a) exists.III.f(x) is continuous at x = a
Solution
The statements about function f can be true or false depending on the specific function. Here's how you can determine the truth of each statement:
I. The limit of f(x) as x approaches a exists: This statement is true if the function approaches a specific value as x gets closer and closer to a from both the left and the right side. If the function approaches different values from the left and the right, or if it doesn't approach any value at all, then the limit does not exist.
II. f(a) exists: This statement is true if the function is defined at x = a. In other words, if you can substitute a into the function and get a real number, then f(a) exists. If substituting a into the function gives you an undefined expression (like dividing by zero), then f(a) does not exist.
III. f(x) is continuous at x = a: This statement is true if both of the above statements are true and the limit of f(x) as x approaches a is equal to f(a). If the limit exists and f(a) exists, but they are not equal, then the function is not continuous at x = a. If either the limit does not exist or f(a) does not exist, then the function is not continuous at x = a.
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(x)=⎧⎩⎨⎪⎪⎪⎪|x+1|,1,x+1√−1x,x<0x=0x>0.𝑓(𝑥)={|𝑥+1|,𝑥<01,𝑥=0𝑥+1−1𝑥,𝑥>0. Which of the following is/are incorrect?Question 1Answera.f(x)𝑓(𝑥) is continuous at x=0𝑥=0.b.f(x)𝑓(𝑥) is continuous for all real numbers x𝑥. c.f(x)𝑓(𝑥) is differentiable at x=0𝑥=0.d.f(x)𝑓(𝑥) is differentiable at all real numbers x𝑥.
Select the correct answer.Consider function f.𝑓(𝑥)={2𝑥,𝑥<0-𝑥2−4𝑥+1,0<𝑥<212𝑥+3,𝑥>2Which statement is true about function f? A. The function is continuous. B. The function is increasing over its entire domain. C. The domain is all real numbers. D. As x approaches positive infinity, 𝑓(𝑥) approaches positive infinity.
If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?I. g(x) = 1 at least onceII. lim𝑥→3.5𝑔(𝑥)=𝑔(3.5)III. lim𝑥→3−𝑔(𝑥)=lim𝑥→3+𝑔(𝑥) I. g(x) = 1 at least onceII. x→3.5lim g(x)=g(3.5)III. x→3−lim g(x)= x→3+lim g(x)
b) f(x) = 3𝑥2; 𝑑𝑓𝑑𝑥(1)= therefore the function is
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