Jacob Scott's Review Sheet for Midterm 1
Practice Midterm
Problem 1:
1. Define the limit of a sequence. (In other words, write " limn→∞an = L means ....")
2. Prove that limn→∞(n/n-1) = 1.
3. Prove that limn→∞(1/n2+sin n) = 0.
Problem 2:
1. Define the limit of a function at ∞. (In other words, write " limx→∞f(x) = L means ....")
2. Prove that limx→∞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 limx→3-√(3 - x) = 0.
2. Prove that limx→4x2 - x + 1 = 13.
3. Prove that limx→1/2(1-x/1+x) = 1/3.
4. Prove that limx→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 limx→af(x) = ∞.
2. Write down a definition for limx→-∞f(x) = -∞.
3. Briefly explain in words what your definitions are doing.
4. Use your definition to prove that limx→3(1/(x-3)2) = ∞.
Problem 6:
1. State the various limit laws. Be explicit about when they apply (e.g., does limx→af(x)g(x) = limx→af(x) · limx→ag(x) if limx→af(x) or limx→ag(x) do not exist?).
2. Prove that limx→a-6f(x) = -6 limx→af(x).
3. Prove via the definition of limit that if limx→a^+f(x) = L and limx→a^-f(x) = L, then limx→af(x) = L.
Problem 7:
1. State the Squeeze Theorem.
2. Prove the Squeeze Theorem.
3. Prove that limx→0x2sin(e1/x) = 0.
4. Why could you NOT solve this problem with the product limit law, e.g. limx→0x2sin(e1/x) ≠ limx→0x2 · limx→0sin(e1/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 limx→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) = ex - 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. limx→1(x2-1/|x-1|)
2. limx→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. limx→∞(3000x2-2x+7/x3-x+1)
4. limx→-∞ x - (x/1+(1/x))
5. limt→1(3-√(t+8)t-1
6. limx→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
![320_Figure.png](https://secure.tutorsglobe.com/CMSImages/320_Figure.png)
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) limx→1f(x) = 2
(c) limx→3^+f(x) = 6
(d) limx→3^-f(x) = -1
(e) f(3) = 0
(f) limx→-4f(x) = 5
(g) f is discontinuous at x = -4
(h) limx→7f(x) = ∞
(i) f(7) = 2
(j) limx→∞f(x) = 1
(k) limx→-∞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)?