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2. Fill in the blank with "all", "no", or "some" to make the following statements true. • If your answer is "all", explain why. • If your answer is "no", give an example and explain. • If your answer is "some", give two examples that demonstrate when the statement is true and when it is false. Explain your examples. Note: An example must include either a graph or a specific function. (a) For ________ functions f and g, if lim_{x?a} g(x) = 0, then lim_{x?a} f(x)/g(x) does not exist. (b) For ________ functions g, if lim_{x?a^+} g(x) = 2 and lim_{x?a^-} g(x) = -2, then lim_{x?a} (g(x))^2 exists. (c) For ________ functions f and g, if lim_{x?a} f(x) exists and lim_{x?a} (f(x) + g(x)) exists, then lim_{x?a} g(x) exists. (d) For real values of x, the functions f(x) = (x^2 - 9)/(x - 3) and g(x) = x + 3 are equal. (e) For real numbers a, lim_{x?a} f(x) = f(a)

          2. Fill in the blank with "all", "no", or "some" to make the following statements true.
• If your answer is "all", explain why.
• If your answer is "no", give an example and explain.
• If your answer is "some", give two examples that demonstrate when the statement is true and when it is false. Explain your examples.
Note: An example must include either a graph or a specific function.
(a) For ________ functions f and g, if lim_{x?a} g(x) = 0, then lim_{x?a} f(x)/g(x) does not exist.
(b) For ________ functions g, if lim_{x?a^+} g(x) = 2 and lim_{x?a^-} g(x) = -2, then lim_{x?a} (g(x))^2 exists.
(c) For ________ functions f and g, if lim_{x?a} f(x) exists and lim_{x?a} (f(x) + g(x)) exists, then lim_{x?a} g(x) exists.
(d) For ________ real values of x, the functions f(x) = (x^2 - 9)/(x - 3) and g(x) = x + 3 are equal.
(e) For ________ real numbers a, lim_{x?a} f(x) = f(a)

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2. Fill in the blank with "all", "no", or "some" to make the following statements true.
• If your answer is "all", explain why.
• If your answer is "no", give an example and explain.
• If your answer is "some", give two examples that demonstrate when the statement is true and when it is false. Explain your examples.
Note: An example must include either a graph or a specific function.
(a) For functions f and g, if limx?a g(x) = 0, then limx?a f(x)/g(x) does not exist.
(b) For functions g, if limx?a^+ g(x) = 2 and limx?a^- g(x) = -2, then limx?a (g(x))^2 exists.
(c) For functions f and g, if limx?a f(x) exists and limx?a (f(x) + g(x)) exists, then limx?a g(x) exists.
(d) For real values of x, the functions f(x) = (x^2 - 9)/(x - 3) and g(x) = x + 3 are equal.
(e) For real numbers a, limx?a f(x) = f(a)

Added by Nicol-S N.

Calculus: Early Transcendentals

James Stewart 8th Edition

Thanks for your feedback!

Fill in the blank with "all", "no", or "some" to make the following statements true. If your answer is "all", explain why. If your answer is "no", give an example and explain. If your answer is "some", give two examples that demonstrate when the statement is true and when it is false. Explain your examples. Note: An example must include either a graph or a specific function. (a) For ______ functions f and g, if lim_{x→a} g(x) = 0, then lim_{x→a} f(x)/g(x) does not exist. (b) For ______ functions g, if lim_{x→a^+} g(x) = 2 and lim_{x→a^-} g(x) = -2, then lim_{x→a} (g(x))^2 exists. (c) For ______ functions f and g, if lim_{x→a} f(x) exists and lim_{x→a} (f(x) + g(x)) exists, then lim_{x→a} g(x) exists. (d) For ______ real values of x, the functions f(x) = (x^2 - 9)/(x - 3) and g(x) = x + 3 are equal. (e) For ______ real numbers a, lim_{x→a} f(x) = f(a)

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0:00 Let me give you a few examples.

00:02 The first one is going to be sum.

00:05 And so let me circle that and give you one that works.

00:08 So maybe the one in works, i'll put in.

00:12 So what makes that statement true would be a function that would be as simple as asking you to do the limit as x approaches 0 of 1 over x.

00:23 That limit does not exist, and that's partly because the limit is x approaches 0 of x equals 0.

00:28 But a counterexamination, so what makes it not true is a limit as x approaches zero and it's a pretty common one.

00:39 So that's why i'm choosing it's sine of x over x.

00:42 So it has the same dilemma as up here, but this limit actually equals 1.

00:47 So let's look at b where the way i'm thinking about this is that this is true for all.

00:56 So let me circle that and the explanation is, if you did the left -hand limit as x approaches to on the left side of g of x squared that would equal negative two squared which equals four and then i would do the right -hand limit to on the right side of g of x squared which would be positive two squared which equals four and since four equals four well technically i'm saying this equals this but i'm going to write it slower that means the limit as x approaches a wait is it a yeah i should have wrote a here and a here of g of x squared exist and it equals four so therefore it exists so that should be true for all at least my understanding so the next one i would go ahead and commit to saying this is true for all because i'm drawing a blank on where a situation where that wouldn't be true.

02:28 And i would recall that, well, i don't know if it's a definition or if it's a property.

02:33 So may i'll call it the property of limits.

02:37 You know, it might even be the addition property of limits, maybe even the subtraction property of limits.

02:45 So that's how i know that's true...

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