1. Use a graph to estimate the limit. Use radians unless degrees are indicated by θ°. (Round your answer to four decimal places.)

Question

lim
θ → 0 θ/tan(7θ)

2. Assuming that limits as x → ∞ have the properties for limits as x → c, use algebraic manipulations to evaluate lim x → ∞ for the function.

f(x) = x − 8/8 + 5x2

3. Same thing as 2 but
f(x) = x4 + 4x/x4 + 3x5

4.Same thing again but
f(x) = 5e−x + 4/4e−x + 5

Please help thanks

Answer 1

#1 You know that as x->0
sinx/x -> 1
tanx/x -> 1
You have
θ/tan7θ = (1/7) (7θ/tan7θ) -> (1/7)(1) = 1/7
As the graph will show
http://www.wolframalpha.com/input/?i=plot+y%3Dx%2F(tan7x),+-0.3+%3C%3D+x+%3C%3D+0.3

#2 some parentheses would help
f(x) = (x − 8)/(8 + 5x2)
divide by x^2
f(x) = (1/x - 8/x^2)/(5 + 8/x^2)
as x->∞ f(x) -> 0/5 = 0

#3
f(x) = (x^4 + 4x)/(x^4 + 3x^5)
lim = 0 again, for the same reason

#4
f(x) = (5e^−x + 4)/(4e^−x + 5)
as x->∞ e^-x -> 0, so
f(x) -> (0+4)/(0+5) = 4/5