Limit Law
Evaluate the limits using basic limit
Limx→2(x)
Limx→2(5)
Limx→2(x) = 2
Limx→2(5) = 5
Name the three types of discontinuities
removable, jump, infinite
limx→-1g(x)=??
limx→-1g(x)=3
Write the point-slope form equation of the line that passes through:
(-10, 3) and (23, 4).
y = (1/33)x + 109/33
What is the definition of an inverse function? (formula)
f-1(f(x)) = x
Evaluate Limx→-2(3x3-2x+7)
Limx→-2(3x3-2x+7) = 3(-2)3 - 2(-2) + 7 = 13
What are the three conditions that continuity requires
f(a) exists, Limx→af(x) exists, Limx→af(x)=f(a)
Slide 3
Limx➝2-f(x) = 8
Limx➝2+f(x) = 3
Limx➝2f(x) = DNE
Write the following equation in equivalent exponential form.
log (100) = 2
102 = 100
How can you determine if a function is even or odd?
f(-x) = f(x) means even; f(-x) = -f(x) means odd
Write out the limit laws to: sum, difference, constant multiple, product, quotient, power, and root in regards to Limx→a of f(x) and g(x)
slide 2
determine whether f(x)=x+2x+1 is continuous at x= -1, classify the continuity
f(x)=x+2x+1 → f(-1)=-1 +2-1 +1 = 10, undefined, therefore discontinuous
Limx→-1- f(x)= - infinity and Limx→-1+ f(x)= + infinity, infinite discont.
Find the following (slide 7)
Limx→ -2- f(x); Limx→ -2+ f(x); Limx→ -2 f(x)
Limx→ 2- f(x); Limx→ 2+ f(x); Limx→ 2 f(x)
Slide 8
Which of the following is not a trig identity?
a) 1 + tan^2(x) = sec^2(x)
b) 1 + cot^2(x) = csc^2(x)
c) tan^2(x) - 1 = sec^2(x)
d) tan(x) = sin(x)/cos(x)
C
Slide 14
Slide 15
Use limit laws to evaluate Limx→6(2x-1)(x+4)), indicating the limit law
1) product: Limx→6(2x-1) x Limx→6(x+4))
2) difference and root: (Limx→6(2x) - Limx→6(1)) x (Limx→6(x+4))
3) constant and sum: (2Limx→6(x) - Limx→6(1)) x (Limx→6(x) + Limx→6 (4))
Evaluate: (2(6)-1) x ((6) + 4)
Limx→6(2x-1)(x+4)) = 1110
Take the intervals over which the function:
f(x)=x-1x2+2xis continuous
A= x2+2x; x= -2 and 0, discontinuous at those points b/c you can’t have #/0, thus continuous intervals are (-∞, -2) U (-2,0) U (0, ∞)
Slide 9
(slide 10)
Limx→ -2 [2f(x) + g(x)]
Limx→ -2 [2f(-2) + g(-2)] = (2)(3) + (0) = 6
Limx→ -2 [f(x) x g(x)]
Limx→ -2 [f(-2) x g(-2)] = (3)(0) = 0
Limx→ -2 [g(f(x))]
Limx→ -2 [g(f(-2))] = Limx→ -2 g(3) = 2
Limx→ 0
g(x)= DNE
Find the amplitude, period and phase shift with direction for the following function: y=-3+4sin(2x+π)
Amplitude: 4; period: π; phase shift: left π, down 3
Slide 16
Slide 12
Apply squeeze theorem to evaluate Limx→0(xcosx)
Limx→0(xcosx)=0 (full workup is in slide 1)
Sketch a graph of the function y = f(x) with the following:
Domain of f is [0,5]
Limx→1+f(x) and the Limx→1-f(x) exists and are equal
f(x) is left continuous but not continuous at x=2, and right continuous but not continuous as x=3
f(x) has a removable discontinuity at x=1, a jump discontinuity at x=2, and the following limits hold:
Limx→3-f(x)= -∞
Limx→3+f(x)= 2
Slide 6
Sketch a graph with the following properties
Limx→2f(x)=1, Limx→4-f(x)=3), Limx→4+f(x)=6, f(4) is not defined
Slide 5
Verify that the following equation is an identity.
sinx/(sinx + cosx)= (tanx)/(1 + tan x)
Slide 13
If f(x) is continuous over [0,2], f(0)>0 and f(2)>0, can we use IVT to conclude that f(x) has no zeros in the interval [0,2]? Explain.
Slide 11