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Question 5: Does the limit 1+n lim n-00 n exist? If so, what is its value? If not, why not?

Question 5: Does the limit 1+n lim n-00 n exist? If so, what is its value? If not, why not?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Now we can write question 2 above in a more compact form. If the limit 1 lim n00 n has a value, then it is the value that should be put in the box for question 2. Question 4: Is this compact form (above), which clearly uses the symbol for infinity (in the "n → 00" part), consistent with what we understand to have been Aristotle's concept of infinity? Question 5: Does the limit 1+n lim n-00 n exist? If so, what is its value? If not, why not? Question 6: Does the limit (1+h)² – 1? lim h→0 exist? If so, what is its value? If not, why not? Optional Question 7 [only for bonus points]: Is it true that 3 x 3 x lim y→0_4x+ 5 y lim lim lim T0 4x +5 y Why? B回 回ロ凸 86% CH W 有道
Transcribed Image Text:Now we can write question 2 above in a more compact form. If the limit 1 lim n00 n has a value, then it is the value that should be put in the box for question 2. Question 4: Is this compact form (above), which clearly uses the symbol for infinity (in the "n → 00" part), consistent with what we understand to have been Aristotle's concept of infinity? Question 5: Does the limit 1+n lim n-00 n exist? If so, what is its value? If not, why not? Question 6: Does the limit (1+h)² – 1? lim h→0 exist? If so, what is its value? If not, why not? Optional Question 7 [only for bonus points]: Is it true that 3 x 3 x lim y→0_4x+ 5 y lim lim lim T0 4x +5 y Why? B回 回ロ凸 86% CH W 有道
Question 1: The concept of infinity which is most often used in applications of mathematics to real world problems is usually attributed to the ancient Greek philosopher Aristotle. What was this concept of infinity? Is it consistent with the fact that infinity is not now usually considered to be a real number? Question 2: "If n increases without end, 1/n becomes closer and closer to Does this make sense? If so, please fill in the box. Otherwise, explain the problem! Question 3: "If the positive natural number n (i.e. n is one of 1, 2, 3 and so on) increases without end, (-1)" becomes closer and closer to ." Does this make sense? If so, please fill in the box. Otherwise, explain the problem! [ Note that "(-1)" " means (-1) × ... × (-1), where the “(-1)" appears n times in total. ] You will have noticed that these questions are very "wordy", but somehow always have basically the same form. Mathematicians like to invent symbols or ways of writing which can replace long stretches of text. We can write f(x) to represent a formula which depends upon x, and lim f(x) to mean "the value, that f(x) gets ever closer to, as x gets ever closer to, but not equal to, a". The somewhat frustrating aspect of this is that we have to add "if it exists"! If it does exist, then its value is called "the limit of f(x) as x approaches a". Finally, the letters "f", "x" and "a" are just names, in the sense of Shakespeare's "A rose by any other name would smell as sweet". Now we can write question 2 above in a more compact form. If the limit lim n 00 n has a value, then it is the value that should be put in the box for question 2. Question 4: Is this compact form (above), which clearly uses the symbol for infinity (in the
Transcribed Image Text:Question 1: The concept of infinity which is most often used in applications of mathematics to real world problems is usually attributed to the ancient Greek philosopher Aristotle. What was this concept of infinity? Is it consistent with the fact that infinity is not now usually considered to be a real number? Question 2: "If n increases without end, 1/n becomes closer and closer to Does this make sense? If so, please fill in the box. Otherwise, explain the problem! Question 3: "If the positive natural number n (i.e. n is one of 1, 2, 3 and so on) increases without end, (-1)" becomes closer and closer to ." Does this make sense? If so, please fill in the box. Otherwise, explain the problem! [ Note that "(-1)" " means (-1) × ... × (-1), where the “(-1)" appears n times in total. ] You will have noticed that these questions are very "wordy", but somehow always have basically the same form. Mathematicians like to invent symbols or ways of writing which can replace long stretches of text. We can write f(x) to represent a formula which depends upon x, and lim f(x) to mean "the value, that f(x) gets ever closer to, as x gets ever closer to, but not equal to, a". The somewhat frustrating aspect of this is that we have to add "if it exists"! If it does exist, then its value is called "the limit of f(x) as x approaches a". Finally, the letters "f", "x" and "a" are just names, in the sense of Shakespeare's "A rose by any other name would smell as sweet". Now we can write question 2 above in a more compact form. If the limit lim n 00 n has a value, then it is the value that should be put in the box for question 2. Question 4: Is this compact form (above), which clearly uses the symbol for infinity (in the
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