Finding Limits of Sums of Terms w/ Diff Limits

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Discussion Overview

The discussion revolves around the limits of sums of terms with different limits as the variable t approaches infinity. Participants explore the behavior of sequences where one term tends to infinity while another tends to zero, and the implications of their coefficients on the overall limit.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the limit of the sum of two terms, where one term approaches zero and the other approaches infinity, is infinity, based on the coefficients of t in each term.
  • Another participant argues that the coefficients do not matter, stating that if one sequence diverges to infinity while another converges to zero, the sum will still diverge to infinity.
  • A later reply reiterates the point about coefficients, emphasizing that the limit behavior remains consistent regardless of their values.

Areas of Agreement / Disagreement

There is disagreement regarding the relevance of the coefficients of t in determining the limit of the sum. Some participants maintain that coefficients are irrelevant, while others suggest they may play a role under certain conditions.

Contextual Notes

The discussion does not resolve the implications of coefficients on the limit, and the mathematical steps regarding the limits of sequences are not fully explored.

Sun God
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For instance:

1/(e^.5t) + 1/(e^-7t)

As t grows larger, the left term goes to 0, but the right term goes to infinity.

Would I be correct in saying that the limit of the sum is infinity because the (absolute value of the) coefficient of t in the term that tends to infinity is larger than the coefficient of t in the term that tends to 0?

What if both t's had the same coefficient?
 
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It doesn't matter what the coefficients are. If \{a_n\} and \{b_n\} are any two sequences such that \lim_{n\to\infty} a_n= \infty and \lim_{n\to\infty}b_n= 0, then \lim_{n\to\infty} (a_n+ b_n)= \infty.

More generally, if \lim_{n\to\infty} a_n= A and \lim_{n\to\infty}b_n= 0, then \lim_{n\to\infty} (a_n+ b_n)= A.
 
Last edited by a moderator:
Fixed you LaTeX script. There were [ itex] tags mixed in with [ math] tags.
HallsofIvy said:
It doesn't matter what the coefficients are. If \{a_n\} and \{b_n\} are any two sequences such that \lim_{n\to\infty} a_n= \infty and \lim_{n\to\infty}b_n= 0, then \lim_{n\to\infty} (a_n+ b_n)= \infty.

More generally, if \lim_{n\to\infty} a_n= A and \lim_{n\to\infty}b_n= 0, then \lim_{n\to\infty} (a_n+ b_n)= A[/itex].
 
Thanks, Mark
 

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