# 137_PS3_Limits.pdf - Problem Sets u2013 Limits and...

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Problem Sets – Limits and continuityThese problem sets contain additional practice problems that correspond to the lecture material. There are hundredsmore in the calculus textbooks on reserve at the library, or on the Davis shelves close to QA303.Limits1. Evaluate the following limits.a)limx0+x-1x-1b)limx→∞sinxxc) limx0|x|xContinuity2. Thesign function, denoted by sgn(x), is defined by,sgn (x) =1,x >00,x= 0-1,x <0a) Sketch sgn(x), indicating any discontinuities.b) Sketchf(x) = sgn(x+ 1)-sgn(x-1), indicating any discontinuities.3. Determine the values of the real numbers{A, B, C}that make the following piece-wise defined functions continuousfor allx.f(x) =A3exp [x-1]Ax2-B2-Bxx11< x <2x2(here, exp[x] is the exponentialex).4.Supposefandgare continuous on [a, b] and thatf(a)< g(a), butf(b)> g(b).Prove thatf(x) =g(x)for somexin [a, b].Hint:Your proof should be very short.More interesting examples
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