indeterminate forms

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types of indeterminate forms:

  1.    [latex]\frac00[/latex]

  2.    [latex] \frac\infty\infty [/latex]

  3.    [latex] 0\;\times\infty [/latex]

  4.    [latex] \infty-\infty [/latex]

  5.    [latex] 1^\infty [/latex]

  6.    [latex] 0^0[/latex]

  7.    [latex] \infty^\infty [/latex]


This all types of limits can be evaluated by using the L' Hospital's rule.
  L' Hospital's rule

If f(x) and g(x) are two function of x which can be expanded by Taylor's series in the neighborhood of x = a and if  [latex] \lim_{x\rightarrow a}f\left(x\right)=f\left(a\right)=0 [/latex],  [latex] \lim_{x\rightarrow a}g\left(x\right)=g\left(a\right)=0 [/latex], then

[latex]\lim_{x\rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\lim_{x\rightarrow a}\frac{f'\left(x\right)}{g'\left(x\right)}[/latex]

 
formulas:

[latex]\lim_{x\rightarrow 0}\frac{\sin x}x=1[/latex]