Answer :
Final answer:
This response clarifies the concept of limits in calculus, specifically addressing common misconceptions and explaining true and false statements by providing relevant examples.
Explanation:
This series of questions is all about the concept of limits in calculus.
(a) False. The limit of a function as x approaches c doesn't require f to be defined at x=c. Example: f(x)=(x^2-1)/(x-1) for x≠1.
(b) False. Sometimes you can find limx→cf(x) by finding f(c) but not always. Example: f(x) = (x^2-4)/(x-2) for x≠2, f(2) is undefined but the limit as x approaches 2 is 4.
(c) False. The limit of a function as x approaches c may not be equal to f(c). For example: f(x) = x when x≠2 , f(2) = 10, but limx→2f(x) = 2.
(d) False. If f(c)=10 it doesn't mean that limx→cf(x)=10. Take f(x) = x when x≠2, f(2)=10, but the limit as x approaches 2 is 2.
(e) False. A function can have at most two limits (from right and left) as x approaches a given point.
(f) False. If limx→4f(x)=10, then we can make f(x) close to 10 by selecting x sufficiently close to 4, not 10.
(g) True. If limx→6f(x)=[infinity], we can make f(x) become as large as we like by choosing values of x close to 6 which is the definition of limit approaching infinity.
(h) True. If limx→[infinity]f(x)=100, we can find values of f(x) close to 100 by selecting sufficiently large x-values.
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