Limit of Indeterminate Form 1 raised to infinity


The limit laws of calculus are usually fairly easy to use as long as you can apply them, but there are many instances where the limit laws are not applicable.

The following limit of (x divided by (x-3)) raised to the x power is of the form one raised to the infinity. What follows is a step by step solution that allows us to use L’Hospital’s rule for limits involving 0/0 or infinity/infinity.

Logarithm functions come in handy when dealing with or trying to solve exponential equations or expressions, so if we take the natural log of both sides the exponent x becomes a factor and after some algebra which involves dividing by the reciprocal we get a quotient that is in the form 0/0.

This means we can take the derivative of the numerator and the denominator and then take the limit again(this is the step where L’Hospital’s rule is actually used) obtaining a limit of 3. Since we took the natural log at the beginning we have to exponentiate( e both sides) to get the sought after limit which is e raised to the third power.

Difficult L’Hospital limit of the form one raised to infinity

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