Worksheet 2 with Solutions PDF

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Worksheet 2 with Solutions...

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Calculus and its Applications 2019-20 Week 2 Workshop, January 21, 2020 Introduction to infinite limits We can make sense of infinite limits if we replace “arbitrarily close to L” by “arbitrarily large” in the definition of limits; see Section 1.6: Definition 1. We say that limx→a f (x) = +∞ if we can make f (x) arbitrarily large by forcing x to be sufficiently close to a, but not equal to a. Similarly, a limit can be −∞ if we can make f (x) arbitrarily small by forcing x to be sufficiently close to a. These definitions also apply to one-sided limits. Simple examples of infinite limits are as follows: 1 = +∞ and |x| 1 lim = −∞ and x→0− x lim

x→0

1 = +∞, x2n 1 lim = +∞. x→0+ x lim

x→0

Question 1. Sketch the graphs of the functions in the above examples, setting n = 1. Solution While these graphs are standard, you should sketch them to “see” what the limits are supposed to be. Question 2. How close does x have to be to 0 in order for f (x) = 10000; larger than M ?

1 |x|

to be larger than 100; larger than

Solution While we will not use the formal ǫ/δ-definition of limits, you should be able to argue along the lines of “if we 1 1 require |x| < 100 , then |x| > 100”, and so forth. Question 3. Determine the following limits: lim

x→3−

x2 − 2x − 1 , x−3

lim

x→3+

x2 − 2x − 1 , x−3

and

lim

x→3+

x2 − 2x − 3 . x−3

Solution We calculate x2 − 2x − 1 x2 − 2x − 1 = −∞, lim+ = +∞, x−3 x−3 x→3 (x − 3)(x + 1) x2 − 2x − 3 = 4. = lim lim+ x−3 x−3 x→3+ x→3 lim

x→3−

and

Question 4. The function f satisfies limx→7− f (x) = −∞. What can you conclude about the graph of f ? Solution Any qualitative description is fine; you should have encountered vertical asymptotes in your reading. We can also define “limits at infinity” by replacing “sufficiently close to a” by “sufficiently large”: 1

Definition 2. We say that limx→+∞ f (x) = L if we can force f (x) to be arbitrarily close to L by forcing x to be a sufficiently large positive number. Similarly, we have Definition 3. We say that limx→−∞ f (x) = L if we can force f (x) to be arbitrarily close to L by forcing x to be a sufficiently large negative number. Question 5. Let f (x) = 2 + x12 ; explain why limx→+∞ f (x) = 2. How large does x have to be if we want f (x) to be within ±0.01 of 2; within ±0.0001? Solution You need to take x > 10 and x > 100, respectively. Question 6. For which values of a do we have limx→+∞ xa = 0? Solution Clearly, a < 0 must hold. All Limit Laws – such as the Sum Law, the Product Law, the Quotient Law, etc. – hold equally for limits at infinity. Use these Laws to evaluate the following limits. Question 7. Calculate 3+ x→+∞ 5 + lim

2 x 4 x

,

lim

x→+∞

3x + 2 , 5x + 4

lim

5x2 + 3x + 5 √ , + 100x + x

x→+∞ x2

and lim

x→+∞

x2 + 200x + 5 . x3 + x + 1

Solution We calculate lim

x→+∞

3+ 5+

2 x 4 x

3 = . 5

lim

x→+∞

3x + 2 3 = , 5x + 4 5

5x2 + 3x + 5 √ = 5, x→+∞ x2 + 100x + x lim

2

x2 + 200x + 5 = 0. x→+∞ x3 + x + 1

and lim

Extra questions (Only attempt these after you have finished with the other questions.) 1 Question 8. Show that, for 0 < x < M and M > 1π , we have f (x) = sin1 x > M . Use this result to determine limx→0+ f (x), limx→0− f (x), and limx→0 f (x), and to justify your answer.

Solution You should be familiar with sin x < x for x > 0, both from the textbook and from Lectures, which you can use without proof. Hence, f (x) > 1x > M , as required. It follows that limx→0+ f (x) = +∞; the fact that f (x) is odd, due to sin being an odd function, then implies limx→0− f (x) = −∞. Since the one-sided limits are not equal, limx→0 f (x) does not exist. Question 9. How would you define limx→+∞ f (x) = +∞? Using your definition, show that limx→+∞ ln x = +∞. (How large do you have to force x to be in order to achieve ln x > M ?) Solution You could say: limx→+∞ f (x) = +∞ if and only if you can make f (x) arbitrarily large by making x sufficiently large. In order to show that limx→+∞ ln x = +∞, you can argue as follows: for any M , no matter how large, we have that ln x > M if we take x > eM . Question 10. If limx→a f (x) = +∞ and limx→a g(x) = 0, what can you say about limx→a f (x)g (x)? Solution Nothing can be said in general: set a = 0 and look at the various combinations of f (x) = |x|−1 or f (x) = x−2 and g(x) = x or g(x) = x2 . Question 11. Recall the precise definition of a limit, which is also known as the ǫ/δ definition: Let f be some function defined on an open interval containing a, except possibly at a itself. Then, we write limx→a f (x) = L if and only if, for every number ǫ > 0, there exists a number δ > 0 such that whenever 0 < |x − a| < δ we have |f (x) − L| < ǫ. Using that definition, prove that limx→4 x2 = 16. Solution Observe that, if |x − 4| < 1, then x ∈ (3, 5), and hence that x + 4 ∈ (7, 9), from which we find |x + 4| < 9. Now, given ǫ > 0, put n ǫo . δ := min 1, 9 For |x − 4| < δ – which, in particular, implies |x − 4| < 1 – we then have ǫ |x2 − 16| = |x − 4| |x + 4| < · 9 = ǫ. 9 Hence, we conclude that limx→4 x2 = 16. Question 12. Attempt Exercises 28, 29, and 31 from Section 1.6. (Find the limits.) • limx→∞

sin2 x , x2

• limx→∞ cos x, √ • limx→∞ (x − x). Solution Since 0 ≤ sin2 x ≤ 1, limx→∞

sin2 x x2

= 0; limx→∞ cos x does not exist; we calculate √ √ √ x(x − 1) (x − x)(x + x) √ √ = +∞. lim (x − x) = lim = lim x→∞ x→∞ x + x→∞ x+ x x 3...