Limits at Infinity With Radicals & Fractional Exponents | Summary and Q&A

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January 25, 2020
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The Organic Chemistry Tutor
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Limits at Infinity With Radicals & Fractional Exponents

TL;DR

The video explains how to solve the limit as x goes to infinity of the x root of x using L'Hopital's rule, with the final answer being one.

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Questions & Answers

Q: How does changing the radical into a fractional exponent help in solving the problem?

Changing the radical into a fractional exponent allows us to rewrite the expression as x raised to the 1 over x. This makes it easier to apply logarithms and simplifies the analysis of the limit.

Q: Why do we use logarithms in this problem?

Logarithms help bring the exponent down so that we can use L'Hopital's rule. By taking the natural log of both sides, we can simplify the expression and apply the rule to evaluate the limit.

Q: What is L'Hopital's rule, and why is it used?

L'Hopital's rule states that for an indeterminate form, such as infinity over infinity or zero over zero, the limit can be evaluated by taking the derivative of the numerator and denominator separately. It is used here because direct substitution of infinity over infinity did not yield a determinate result.

Q: How do we convert a logarithmic expression into an exponential expression?

In general, if log base a of b is equal to c, then a raised to the power of c is equal to b. In this case, since the base of a natural log is always e, we can say that e to the 0 power is equal to y, which simplifies to y = 1.

Summary & Key Takeaways

  • The video discusses solving the limit of the x root of x as x approaches infinity.

  • The first step is to set the original expression equal to y and change the radical into a fractional exponent.

  • Using logarithms and L'Hopital's rule, the limit is evaluated as zero, confirming that the answer is one.

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