Some Limits of Indeterminate Forms

March 21, 2008

Evaluate the following limits:

1. \displaystyle\lim\limits_{x\rightarrow 1} \frac{2x^3-3x^2+1}{3x^3-4x^2-x+2}

2. \displaystyle\lim\limits_{x\rightarrow \infty} \big( xe^{-\frac{1}{x}}-x \big)

3. \displaystyle\lim\limits_{x\rightarrow \infty} \frac{\ln(x+e^x)}{3x}
4. \displaystyle\lim\limits_{x\rightarrow 1^+} (2-x)^{\tan(\frac{\pi}{2} x)}
5. \displaystyle\lim\limits_{x\rightarrow 0^+} \frac{1}{\ln(\cos x)} \int_0^x \tan^{-1} t \ dt

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Posted by ldvallejo

Derivatives of Integrals

March 5, 2008

Evaluate the following.

  1. \displaystyle \frac{d}{dx} \int_{x}^{0} \sqrt{1+t^{4}}\ dt
  2. \displaystyle \frac{d}{dx} \int_{\frac{1}{x}}^{\sqrt{x}} \cos t^{2}\ dt,\; x>0
  3. \displaystyle \frac{d^{2}}{dx^{2}} \int_{0}^{x} \left( \int_{1}^{\sin t} \sqrt{1+u^{4}}\ du \right) dt

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Posted by wMw

Locating Extreme Points

March 4, 2008

Find the points of extremum of r(x)={\displaystyle \int_{0}^{x} \frac{\sin t}{t}\ dt} if x>0.

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Posted by wMw