Calculus I, Fall 2013

Quiz 1

Oct 03, 2013 Show all details.

1. Find lim_{θ→0} \frac{sin θ}{tan 2θ}.

2. State the Intermediate Value Theorem (Need not prove). Use it to find a c such that there a root of ”x − 1 = cos x” on (c, c + 1).

3. Write down the definition of lim_{x→2} \frac{1}{x} = 0.5 in terms of ϵ and δ. Then find one δ that works for ϵ = 0.1 (need not be the best δ). Explain.

4. Give precise definitions of the following limits (Just the definition, need not find δ).

(a) lim_{x→c^{−}} f(x) = L (b) lim_{x→−∞} f(x) = ∞

5. Use the ϵ − δ argument to show that if both f(x) and g(x) are continuous at x = c, then so is 2f(x) + 3g(x).

Calculus I, Fall 2013

Quiz 1

Oct 03, 2013 Show all details.

1. Find lim_{θ→0} \frac{sin θ}{tan 2θ}.

2. State the Intermediate Value Theorem (Need not prove). Use it to find a c such that there a root of ”x − 1 = cos x” on (c, c + 1).

3. Write down the definition of lim_{x→2} \frac{1}{x} = 0.5 in terms of ϵ and δ. Then find one δ that works for ϵ = 0.1 (need not be the best δ). Explain.

4. Give precise definitions of the following limits (Just the definition, need not find δ).

(a) lim_{x→c^{−}} f(x) = L (b) lim_{x→−∞} f(x) = ∞

5. Use the ϵ − δ argument to show that if both f(x) and g(x) are continuous at x = c, then so is 2f(x) + 3g(x).