Section 1.2 Limits
Evaluate the following limits. Use limit theorems, not ε - δ techniques. If any of them fail to exist, say so and say why.limx→10x2−100x−10
limx→10x2−99x−10
limx→10x2−100x−9
limx→10f(x), where f(x)=x2 for all x≠10, but f(10)=99.
limx→10√−x2+20x−100
limx→−4x2−16x+4ln|x|
limx→∞x2e4x−1−4x
limx→−∞3x6−7x5+x5x6+4x5−3
limx→−∞5x7−7x5+12x7+6x6−3
limx→−∞2x+3x3x3+2x−1
limx→−∞5x+2x3x3+x−7
limx→∞ax17+bxcx17−dx3, a,b,c,d≠0
limx→∞3x+|1−3x|1−5x
limx→−∞√x6−3√x6+5
limu→∞u√u2+1
limx→∞1+3x√2x2+x
limx→∞√4x2+3x−77−3x
limx→−∞√x2−92x−1
limx→1+√x−1x2−1
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Let f(x)={x2−1|x−1|if x≠1,4if x=1.
Find limx→1−f(x).
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Let F(x)=2x2−3x|2x−3|.
Find limx→1.5+F(x).
Find limx→1.5−F(x).
Does limx→1.5F(x) exist? Provide a reason.
limx→−22−|x|2+x
limx→2−|x2−4|10−5x
limx→4−|x−4|(x−4)2
limx→8(x−8)(x+2)|x−8|
limx→2(1x2+5x+6−1x−2)
limx→−1x2−x−23x2−x−1
limx→16√x−4x−16
limx→83√x−2x−8
limx→42−√x4x−x2
limx→0√1+2x−√1−4xx
Find constants a and b such that limx→0√ax+b−2x=1.
limx→5ex−5√x−1−2
limx→7e√x+2−3x−7
limt→0√sint+1−1t
limx→8x1/3−2x−8
limx→∞(√x2+x−x)
limx→−∞(√x2+5x−√x2+2x)
limx→∞(√x2−x+1−√x2+1)
limx→∞(√x2+3x−2−x)
Is there a number b such that limx→−2bx2+15x+15+bx2+x−2 exists? If so, find the value of b and the value of the limit.
Determine the value of a so that f(x)=x2+ax+5x+1 has a slant asymptote y=x+3.
Prove that f(x)=lnxx has a horizontal asymptote y=0.
Let I be an open interval such that 4∈I and let a function f be defined on a set D=I∖{4}. Evaluate limx→4f(x), where x+2≤f(x)≤x2−10 for all x∈D.
Evaluate limx→1f(x), where 2x−1≤f(x)≤x2 for all x in the interval (0,2).
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Use the squeeze theorem to show that
limx→0x4sin(1x)=0,
limx→0+(√xesin(1/x))=0.
Evaluate the following limits. If any of them fail to exist, say so and say why.
limx→0+[(x2+x)1/3sin(1x2)]
limx→0xsin(ex)
limx→0xsin(1x2)
limx→0√x2+x⋅sin(πx)
limx→0xcos2(1x2)
limx→π/2+xcotx
limx→01−e−x1−x
limx→0e2x−1−2xx2
limx→2ex−e2cos(πx2)+1
limx→1x2−1e1−x7−1
limx→0e−x2cos(x2)x2
limx→1x76−1x45−1
limx→1xa−1xb−1, a,b≠0
limx→0(sinx)100x99sin2x
limx→0x100sin7x(sinx)99
limx→0x100sin7x(sinx)101
limx→0arcsin3xarcsin5x
limx→0sin3xsin5x
limx→0x3sin(1x2)sinx
limx→0sinx√xsin4x
limx→01−cosxxsinx
limθ→3π2cosθ+1sinθ
limx→π2(x−π2)tanx
limx→∞xtan(1/x)
limx→0(1sinx−1x)
limx→0x−sinxx3
limx→0(cscx−cotx)
limx→0+(sinx)(lnsinx)
limx→∞(x⋅lnx−1x+1)
limx→∞ex10x3
limx→∞lnx√x
limx→∞ln3xx2
limx→∞(lnx)2x
limx→1lnxx
limx→0ln(2+2x)−ln2x
limx→∞ln((2x)1/2)ln((3x)1/3)
limx→0ln(1+3x)2x
limx→1ln(1+3x)2x
limθ→π2+ln(sinθ)cosθ
limx→11−x+lnx1+cos(πx)
limx→0(1x2−1tanx)
limx→0+(1x−1ex−1)
limx→0(coshx)1x2
limx→0+(cosx)1x
limx→0+(cosx)1x2
limx→0+xx
limx→0+x√x
limx→0+xtanx
limx→0+(sinx)tanx
limx→0(1+sinx)1x
limx→∞(x+sinx)1x
limx→∞x1x
limx→∞(1+1x)2x
limx→∞(1+sin3x)x
limx→0+(x+sinx)1x
limx→0+(xx+1)x
limx→e+(lnx)1x−e
limx→e+(lnx)1x
limx→0exsin(1/x)
limx→0(1−2x)1/x
limx→0+(1+7x)1/5x
limx→0+(1+3x)1/8x
limx→0(1+x2)3/x
Let x1=100, and for n≥1, let xn+1=12(xn+100xn). Assume that L=limn→∞xn exists. Calculate L.
Find limx→01−cosxx2, or show that it does not exist.
Find limx→2π1−cosxx2, or show that it does not exist.
Find limx→−1arcsinx, or show that it does not exist.
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Compute the following limits or state why they do not exist:
limh→04√16+h−22h
limx→1lnxsin(πx)
limu→∞u√u2+1
limx→0(1−2x)1/x
limx→0(sinx)100x99sin(2x)
limx→∞1.01xx100
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Find the following limits. If a limit does not exist, write 'DNE'. No justification necessary.
limx→0(2+x)2016−22016x
limx→∞(√x2+x−x)
limx→0cot(3x)sin(7x)
limx→0+xx
limx→∞x2ex
limx→3sinx−xx3
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Evaluate the following limits, if they exist.
limx→0f(x)|x| given that limx→0xf(x)=3.
limx→1sin(x−1)x2+x−2
limx→−∞√x2+4x4x+1
limx→∞√x4+2x4−4
limx→∞(ex+x)1/x
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Evaluate the following limits, if they exist.
limx→4[1√x−2−4x−4]
limx→1x2−1e1−x2−1
limx→0(sinx)(lnx)
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Evaluate the following limits. Use “∞” or “−∞” where appropriate.
limx→1−x+1x2−1
limx→0sin6x2x
limx→0sinh2xxex
limx→0+(x0.01lnx)
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Use the ε−−δ definition of limits to prove that
limx→0x3=0. Sketch an approximate graph of f(x)=2x2 on [0,2]. Next, draw the points P(1,0) and Q(0,2). When using the precise definition of limx→1f(x)=2, a number δ and another number ε are used. Show points on the graph which these values determine. (Recall that the interval determined by δ must not be greater than a particular interval determined by ε.)
Use the graph to find a positive number δ so that whenever |x−1|<δ it is always true that |2x2−2|<14.
State exactly what has to be proved to establish this limit property of the function f.
Give an example of a function F=f+g such that the limits of f and g at a do not exist and that the limit of F at a exists.
If limx→a[f(x)+g(x)]=2 and limx→a[f(x)−g(x)]=1 find limx→a[f(x)⋅g(x)].
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If f′ is continuous, use L'Hospital's rule to show that
limh→0f(x+h)−f(x−h)2h=f′(x).Explain the meaning of this equation with the aid of a diagram.