Suppose limx→t−f(x)=limx→t−g(x)lim𝑥→𝑡−𝑓(𝑥)=lim𝑥→𝑡−𝑔(𝑥) and limx→t+f(x)=limx→t+g(x)lim𝑥→𝑡+𝑓(𝑥)=lim𝑥→𝑡+𝑔(𝑥). Then, is limx→tf(x)=limx→tg(x)lim𝑥→𝑡𝑓(𝑥)=lim𝑥→𝑡𝑔(𝑥) true?
Question
Suppose limx→t−f(x)=limx→t−g(x)lim𝑥→𝑡−𝑓(𝑥)=lim𝑥→𝑡−𝑔(𝑥) and limx→t+f(x)=limx→t+g(x)lim𝑥→𝑡+𝑓(𝑥)=lim𝑥→𝑡+𝑔(𝑥). Then, is limx→tf(x)=limx→tg(x)lim𝑥→𝑡𝑓(𝑥)=lim𝑥→𝑡𝑔(𝑥) true?
Solution
Yes, the statement is true. If the left-hand limit and the right-hand limit of the functions f(x) and g(x) are equal as x approaches t, then the overall limit of f(x) and g(x) as x approaches t will also be equal. This is because the overall limit of a function at a point is defined only when the left-hand limit and the right-hand limit exist and are equal. Since both these conditions are satisfied for both f(x) and g(x), we can say that limx→tf(x)=limx→tg(x).
Similar Questions
Assume limx→tf(x)=ylim𝑥→𝑡𝑓(𝑥)=𝑦. Must f𝑓 be defined at x=t𝑥=𝑡?
limx→∞exx
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)
When performing the composition of two functions f(x)𝑓(𝑥) and g(x)𝑔(𝑥), [f∘g](x)[𝑓∘𝑔](𝑥) and [g∘f](x)[𝑔∘𝑓](𝑥) will produce the same answer.Question 5Select one:TrueFalse
Given that lim x→1 f(x) = 1 lim x→1 g(x) = −2 lim x→1 h(x) = 0,find each limit, if it exists. (If an answer does not exist, enter DNE.)(a)lim x→1 [f(x) + 3g(x)] (b)lim x→1 [g(x)]3 (c)lim x→1 f(x) (d)lim x→1 4f(x)g(x) (e)lim x→1 g(x)h(x) (f)lim x→1 g(x)h(x)f(x)
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.