Jacob Scott's Review Sheet for Midterm 1

Practice Midterm

Problem 1:

1. Define the limit of a sequence. (In other words, write " lim_n→∞a_n = L means ....")

2. Prove that lim_n→∞(n/n-1) = 1.

3. Prove that lim_n→∞(1/n²+sin n) = 0.

Problem 2:

1. Define the limit of a function at ∞. (In other words, write " lim_x→∞f(x) = L means ....")

2. Prove that lim_x→∞x√(1/x³+1) = 0. (Hint: if that +1 weren't there, this would be much easier...)

Problem 3:

1. Define the limit of a function at a.

2. Define the right- and left-handed limits of a function at a.

3. How would you change your answer to (1) to get a definition of continuity? Why is that equivalent to the definition of continuity you learned in class?

Problem 4:

Without using limit laws:

1. Prove that lim_x→3-√(3 - x) = 0.

2. Prove that lim_x→4x² - x + 1 = 13.

3. Prove that lim_x→1/2(1-x/1+x) = 1/3.

4. Prove that lim_x→1x³ - 3x + 4 = 2. (If you need help factoring, remember that x - a must be a factor, so divide it out... or just give it to Wolfram Alpha.)

Problem 5:

1. Write down a definition that makes sense for lim_x→af(x) = ∞.

2. Write down a definition for lim_x→-∞f(x) = -∞.

3. Briefly explain in words what your definitions are doing.

4. Use your definition to prove that lim_x→3(1/(x-3)²) = ∞.

Problem 6:

1. State the various limit laws. Be explicit about when they apply (e.g., does lim_x→af(x)g(x) = lim_x→af(x) · lim_x→ag(x) if lim_x→af(x) or lim_x→ag(x) do not exist?).

2. Prove that lim_x→a-6f(x) = -6 lim_x→af(x).

3. Prove via the definition of limit that if lim_x→a^+f(x) = L and lim^{x→a^-}f(x) = L, then lim_x→af(x) = L.

Problem 7:

1. State the Squeeze Theorem.

2. Prove the Squeeze Theorem.

3. Prove that lim_x→0x²sin(e^1/x) = 0.

 4. Why could you NOT solve this problem with the product limit law, e.g. lim_x→0x²sin(e^1/x) ≠  lim_x→0x² · lim_x→0sin(e^1/x)?

Problem 8:

1. State the definition of continuity.

2. Where is f(x) = ⌊x⌋ continuous? Prove it. (That is, prove it is continuous everywhere you say it is, and prove that it isn't continuous everywhere you say it isn't; to do that, find the left- and right-handed limits.)

3. Prove via the definition of limit that lim_x→3⌊x⌋ doesn't exist. Conclude that ⌊x⌋ isn't continuous at x = 3. (In other words, for any potential limit L find an ∈, say ∈ = 0.3 (anything ≤ 1/2 will work) such that there is no δ > 0 such that whenever 0 < |x - 3| < δ, we have |⌊x⌋ - L| < ∈.

Note that this is essentially backwards from a usual limit problem. Also, to be completely formal you will need to use the Triangle Inequality!)

Problem 9:

1. State the Intermediate Value Theorem. (Formally, not via a picture)

2. Illustrate your response to (1), and illustrate why the Intermediate Value Theorem does not hold when f is not continuous (or defined) on the relevant interval.

3. State the Triangle Inequality.

Problem 10:

1. Show that f(x) = 2x⁴ + 7x - 4 has a root between x = -1 and x = 2.

2. Show that f(x) = sin(πx) intersects g(x) = x² somewhere between x = ½ and x = 1.

3. Show that f(x) = e^x - 1/x² has a zero between x = -1 and x = 1.

Problem 11:

Compute the following limits via limit laws, or show that they don't exist (usually by computing left- and right-handed limits).

1. lim_x→1(x²-1/|x-1|)

2. lim_x→0^+x√(1+√(1+√(1+(1/x⁸)))). What if the 8 were replaced by a 7 or 9? What if it were a 2-sided limit (a bit tricky)?

3. lim_x→∞(3000x²-2x+7/x³-x+1)

4. lim_x→-∞ x - (x/1+(1/x))

5. lim_t→1(3-√(t+8)t-1

6. lim_x→3⌈x⌉

Problem 12:

For this problem, you may use any limit laws you wish without explanation.

1. Write down the (limit) definition of derivative.

2. Prove that d/dx(x²) = 2x by the above definition.

3. Prove that d/dx(x/1-x) = 1/(x-1)² by the above definition.

4. Prove that if f(x) and g(x) are differentiable, then d/dx (f(x) + g(x)) = d/dx (f(x)) + d/dx (g(x)).

Problem 13:

1. Let

Where is f continuous? Prove it. (Like above, prove it is continuous everywhere you say it is, and prove that it isn't continuous everywhere you say it isn't.)

Problem 14:

1. Sketch the graph of a single function f(x) with all the following properties:

(a) f has domain R - {1}

(b) lim_x→1f(x) = 2

(c) lim_x→3^+f(x) = 6

(d) lim_x→3^-f(x) = -1

(e) f(3) = 0

(f) lim_x→-4f(x) = 5

(g) f is discontinuous at x = -4

(h) lim_x→7f(x) = ∞

(i) f(7) = 2

(j) lim_x→∞f(x) = 1

(k) lim_x→-∞f(x) = -∞.

Problem 15:

1. Find the slope of the function f(x) = 4√ x at x = 0.

2. Why doesn't that conflict with the definition of function (i.e., the vertical line test)?