Limits (green worksheet)

lim h→0 (f(2+h)-F(2))/(h) =5

I. f is continuous at x=2 II. f is differentiable at x=2 ***THE DERIVATIVE IS NOT CONTINUOUS AT X=2***

lim ((1-cosθ)/(2sin²θ))

¼

picture problem

tbd

lim x→∞ (1+5e∧x)∧(1/x)

e

lim x→a f(x)=L where L is a real number, which of the following must be true?

f'(a) DOES NOT EXIST f(x) is NOT continuous at x=a f(x) is NOT defined at x=a f(a)≠L

If n is a positive integer, then: lim x→+∞ ((xⁿ)/(e^x))

0

lim x→0 (xcscx)

1

a≠0 lim (x²-a²)/(x⁴-a⁴) x→a

1/(2a²)

f(x)=2x²+1 lim x→0 ((f(x)-f(0))/ x²)

2

f(x)= sin x, x<0 x², 0≤x<1 2-x, 1≤x<2 x-3, x≥2 For what values of x is ƒ NOT continuous?

2 only

lim n→∞((3n³-5n)/(n³-2n²+1))

3

lim n→∞((4n²)/(n²+10000n))

4

lim x→0 ((1-cos²(2x))/x²)

4

at x=3 f(x)= x², x<3 6x-9, x≥3

Both continuous and differentiable

lim x→1 ((x)/lnx)

DNE

If ƒ is a continuous function on [a,b], which of the following is necessarily true?

lim x→x₀ f(x)= ƒ(lim x→x₀ x) for x₀ ∈(a,b)

If lim x→3 f(x)=7, which of the following must be true?

none f is not continuous at x=3 f is not differentiable at x=3 f(3)≠7

f(x)= lnx for 0<x≤2 x²ln2 for 2<x≤4 then lim x→2 f(x)

nonexistent

same as above

same