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Question

Graphing Functions, L'Hospital's Rule-

Exercise 1- Give an example of a function f which is continuous everywhere but has a horizontal asymptote. Give an example of a function f which is continuous everywhere but has a slant asymptote.

Exercise 2- Give a proof, using N and δ, of the following fact: If a function f(x) is continuous at x = a, then f(x) cannot have a vertical asymptote.

Exercise 3- Let f(x) = a_nxⁿ + a_n-1x^n-1 + · · · + a₁x + a₀ be a polynomial with n ≥ 1 and a_n ≠ 0. Prove that lim_x→∞ f(x) = ∞ if an > 0 and -∞ if an < 0. Use this to show that lim_x→-∞ f(x) = ∞ if either (i) n is even and a_n > 0 or (ii) n is odd and an < 0; show that lim_x→-∞ f(x) = -∞ if either (iii) n is odd and a_n > 0 or (iv) n is even and a_n < 0.

Exercise 4- Find the limit lim_x→∞(x - ln x).

Exercise 5- Find the limit lim_x→∞ x^1/x.

Exercise 6- Find the limit lim_x→∞(1+ (2/x))^x. More generally, find the limit lim_x→∞(b+(a/x))^x where a is a real number and b is a positive real number.

Exercise 7- Prove that lim_x→∞e^x/xⁿ = ∞ for any positive integer n. (Intuition: The exponential function grows faster than any large power of x.)

Exercise 8- Prove that lim_x→∞ln x/x^p = 0 for any p > 0. (Intuition: The natural logarithm grows slower than any small power of x. Application to computer science: Many sorting algorithms have a worst-case running time of O(n ln n), which is better than O(n^1+p) for any p > 0. But it's worse than O(n). Wait, what?)

Exercise 9- Let A(θ) be the area of the region between the chord PR and the arc PR. Let B(θ) be the area of the triangle PQR. Find lim^{θ→0^+}A(θ)/B(θ).

Exercise 10- Let f and g be functions such that f(x) > 0 for all x. Suppose that lim_x→a f(x) = 0 and lim_x→a g(x) = ∞. Show that lim_x→a(f(x))g(x) = 0. (Hint: This is not an indeterminate form. Intuition: You are multiplying increasingly many copies of increasingly small numbers.)