There are many functions, the graphs of which appear as waves on X-axis of Cartesian plane, with ever increasing amplitude as $x$ approaches $–\infty$. eg. $y=x^n \sin(x)\cos(x)$ Can you provide me with some more examples of such functions?
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Can somebody please fix the formula? This is my first time with MathJax and I'm unable to do so. – Sumant Chopde Aug 27 '18 at 04:59
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1Here's a good MathJax tutorial One thing that helped me a lot is when you see a formula on this site that you don't know how to format, you can right click it it, then choose Show Math as... TeX commands to see how to write it. – saulspatz Aug 27 '18 at 05:04
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@saulspatz Thank you very much. – Sumant Chopde Aug 27 '18 at 05:22
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Functions like$$e^{kx} \sin bx$$ and $$e^{kx} \cos bx$$ with different values of $k$ and $b$ are other examples of wavy functions.
Mohammad Riazi-Kermani
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Can you please provide examples where the amplitude increases as x approaches –∞? – Sumant Chopde Aug 27 '18 at 05:29
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$e^{-2x} cos x$ is such a function. When you let $k<0$ the amplitude goes to infinity as $x\to -\infty$ – Mohammad Riazi-Kermani Aug 27 '18 at 05:33
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@SumantChopde. If you replace $e^{kx}$ with $\cosh(kx) = \frac12(e^{kx}+e^{-kx})$ or $\sinh(kx) = \frac12(e^{kx}-e^{-kx})$ then you get a "wavy function" with an amplitude that tends to infinity at both ends, both as $x \to +\infty$ and as $x \to -\infty.$ – md2perpe Aug 27 '18 at 05:35
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Some examples:
$\frac12(e^x + e^{-x}) \sin x$
$|x| \sin x$
$\ln|x| \sin x$
$\sqrt{1+x^2} \sin x$
$e^{e^x+e^{-x}} \sin x$
md2perpe
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