Indeterminate form: Difference between revisions

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The most common example of an indeterminate form is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by <math>0/0</math>. For example, as <math>x</math> approaches <math>0~</math>, the ratios <math>x/x^3</math>, <math>x/x</math>, and <math>x^2/x</math> go to <math>\infty</math>, <math>1</math>, and <math>0~</math> respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is <math>0/0</math>, which is indeterminate. In this sense, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, by appropriate choices of functions to put in the numerator and denominator. You may in fact find a pair of functions for which the limit is any particular given value. Even more surprising, perhaps, the quotient of the two functions may in fact diverge, and not merely diverge to infinity. For example, <math> x \sin(1/x) / x</math>.

So, giventhe fact that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> bothconverge approachingto <math>0~</math> as <math>x</math> approaches some [[limit point]] <math>c</math>, that fact alone does not give enoughis informationinsufficient forto evaluatingdeterminate the [[limit of a function|limit]]

{{block indent|<math> \lim_{x \to c} \frac{f(x)}{g(x)} .</math>}}