(a) Suppose that a function f(x) is defined for all z in [-1, 1]. Can anything be said about the existence of lim f(x)? Give reasons for your answer. (b) Find the limit of the function 21 + 7 f(x) = 13 – x² + x + 7 as r + 00 and r + -0.
Q: See pencil for correction of problem
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A: We can prove the statement as below using definition of limit
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?1. Consider the function f(x)= {x+2, x>0 -x-6,x<0. Use the table of values to find the limit of the following: a. lim f(x) x→0+= x 0.0001 0.001 0.01 0.1 1 f(x) b. lim f(x) x→0-= x -1.0 -0.1 -0.01 -0.001 -0.0001 f(x) c. Does the limit f(x) x→0 exist? If yes, give its value. If not, why?Find the constant c so that the limit limx->3 (x2+x+c)/(x2-5x+6) exists.
- Let f(x) = (3x^2 + 16x − 12)/(x + 6). Find L such that lim x→−6 f(x) = L, and prove that the limit is correct by completing a δ − ε proof for the limit of the function.If x4 <=ƒ(x)<= x2 for x in [-1, 1] and x2<=ƒ(x) <= x4 for x <-1 and x> 1, at what points c do you automatically know limx-->c ƒ(x)? What can you say about the value of the limit at these points?9. Find the value of a such that exists and find the limit. limx-> -1 (2x2-ax-14)/(x2 -2x-3)
- For the function f(x) (X^2-3 if -3 ≤x2 C) is f continuous at x=2? D) does lim f(x) exist at x->4^-? E) does lim exist f(x) exist at x->4^+?Suppose f : A →R is a function, c is a limit point of A, and limx→c f(x) = L. Prove limx→c kf(x) = kLby using the ε-δ definition of the limit.Find the limit of lim((9+3x)1/2-3)/x as x approaches 0