Indeterminate Limits
Indeterminate limits are mathematical expressions that cannot be assigned a unique value. Although these expressions have no meaning per se, they frequently arise in limit problems in calculus. For example, to compute the derivative (slope) of the sine curve at any point on the curve, one needs to evaluate the limit of sin(x)/x as x approaches 0.
Simple substitution of x = 0 into this expression does not produce a satisfactory result: It gives the indeterminate limit 0/0. But a more careful approach, using the "sandwich principle" of calculus, shows that the correct value of the limit is actually 1.
Generally speaking, a limit of a quotient of two functions, f(x)/ g(x), is said to be indeterminate of the form 0/0 if both f(x) and g(x) approach 0. The actual limit of f(x)/ g(x) in such a case depends on more detailed information about these two functions - intuitively, which one approaches 0 "faster." A powerful tool for evaluating such limits is L'Hospital's Rule, which provides the needed information by comparing the derivatives of f(x) and g(x).
Besides 0/0, the other commonly occurring indeterminate limits are (infinity)/(infinity), (infinity) - (infinity), 1(infinity), (infinity)0, and 00. L'Hospital's Rule applies directly to the first of these; the other four generally need to be transformed algebraically before l'Hospital's Rule can be applied.
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