6.2.25 Evaluate Limit of x to infty

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Discussion Overview

The discussion centers around evaluating the limit of a function involving exponential terms as \( x \) approaches infinity. The focus is on the mathematical reasoning and techniques used to simplify the expression.

Discussion Character

  • Mathematical reasoning, Homework-related

Main Points Raised

  • One participant, DOY357, expresses difficulty with the exponential function in the limit evaluation.
  • Another participant suggests rewriting \( e^{3x} \) as \( t \), indicating that as \( x \) approaches infinity, \( t \) will also approach infinity, which may simplify the problem.
  • DOY357 reiterates the limit expression and presents a step-by-step evaluation, ultimately arriving at a limit of 1.
  • A later reply comments on the interesting nature of the discussion and indicates a willingness to continue with more problems.

Areas of Agreement / Disagreement

There is no explicit consensus on the limit evaluation, but one participant provides a detailed step-by-step approach that leads to a specific result. The initial difficulty expressed by DOY357 suggests uncertainty in the understanding of the concepts involved.

Contextual Notes

The discussion does not address potential assumptions or limitations in the mathematical steps taken, nor does it clarify the definitions of terms used in the limit evaluation.

Who May Find This Useful

Students or individuals seeking assistance with limits involving exponential functions, particularly those who may struggle with the manipulation of such expressions.

karush
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6.2.25 DOY357

Evaluate
$\displaystyle\lim_{x\to \infty}\dfrac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$

sorry I'm clueless with this e stuff:(
 
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If you find e hard, then rewrite $$e^{3x} = t$$. Now t is approaching infinity, and no more e's in this problem.
 
karush said:
6.2.25 DOY357

Evaluate
$\displaystyle\lim_{x\to \infty}\dfrac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$

sorry I'm clueless with this e stuff:(

$\displaystyle \begin{align*} \lim_{x \to \infty} \left( \frac{\mathrm{e}^{3\,x} - \mathrm{e}^{-3\,x}}{\mathrm{e}^{3x} + \mathrm{e}^{-3\,x}} \right) &= \lim_{x \to \infty} \left[ \left( \frac{\mathrm{e}^{3\,x} - \mathrm{e}^{-3\,x}}{\mathrm{e}^{3x} + \mathrm{e}^{-3\,x}} \right) \left( \frac{\mathrm{e}^{3\,x}}{\mathrm{e}^{3\,x}} \right) \right] \\ &= \lim_{x \to \infty} \left( \frac{\mathrm{e}^{6\,x} - 1}{\mathrm{e}^{6\,x} + 1} \right) \\ &= \lim_{x \to \infty} \left( 1 - \frac{2}{\mathrm{e}^{6\,x} + 1} \right) \\ &= 1 - 0 \\ &= 1 \end{align*}$
 
well that was interesting a very helpfull
I have a few more coming up but will start a new post
 

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