⧼exchistory⧽
13 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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What is the domain and the range of each one of the six trigonometric functions?
BBy Bot
Nov 03'24
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[/math]
Evaluate each of the following indeterminate forms (see Problem):
- [math]\lim_{x\goesto0+} (\sin x)^{\tan x}[/math]
- [math]\lim_{x\goesto0+} x^{1-\cos x}[/math].
BBy Bot
Nov 03'24
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[/math]
If [math]\lim_{x\goesto a} f(x) = \lim_{x\goesto a} g(x) = \pm \infty[/math], it is not immediately apparent whether or not [math]\lim_{x\goesto a} (f(x)-g(x))[/math] exists. Such limits are commonly called indeterminate forms of the type [math]\infty - \infty[/math]. The usual method of evaluation is to express the difference [math]f(x) - g(x)[/math] as a quotient and then to try to find its limit. For example, we write
[[math]]
\frac{e^x}{e^x-1} - \frac1x =
\frac{xe^x - (e^x-1)}{x(e^x - 1)}
,
[[/math]]
and, as [math]x[/math] approaches zero, the limit of the right side can be obtained by two applications of L'H\^opital's Rule. Evaluate
- [math]\lim_{x\goesto0} \left(\frac{e^x}{e^x-1} - \frac1x\right)[/math]
- [math]\lim_{x\goesto0} \left[\frac{(x^2+8)^{\frac13}} {2x^2} - \frac1{x^2}\right][/math]
- [math]\lim_{x\goesto0} \left(\frac{x^2+3x+5} {\sin x} - \frac5x\right)[/math]
- [math]\lim_{t\goesto0} \left(\cot t - \frac{1-2t}t\right)[/math]
- [math]\lim_{x\goesto0+} \left(\frac1x + \ln x\right)[/math]
- [math]\lim_{x\goesto\frac{\pi}2} (\sec x - \tan x)[/math].