Maths-
General
Easy

Question

Lt subscript x not stretchy rightwards arrow 0 end subscript open parentheses 1 plus fraction numerator 2 x over denominator 3 end fraction close parentheses to the power of 1 over x end exponent

  1. e to the power of 2 over 3 end exponent   
  2. e to the power of 1 third end exponent   
  3. e to the power of 3 over 2 end exponent   
  4. e cubed   

hintHint:

Using L'Hospital's Rule, solve the sum.

The correct answer is: e to the power of 2 over 3 end exponent 


    L e t space y equals limit as x rightwards arrow 0 of open parentheses 1 plus fraction numerator 2 x over denominator 3 end fraction close parentheses to the power of 1 over x end exponent
A p p l y i n g space log space o n space b o t h space t h e space s i d e s comma
ln y equals ln open square brackets limit as x rightwards arrow 0 of open parentheses 1 plus fraction numerator 2 x over denominator 3 end fraction close parentheses to the power of 1 over x end exponent close square brackets
ln y equals limit as x rightwards arrow 0 of open square brackets ln open parentheses 1 plus fraction numerator 2 x over denominator 3 end fraction close parentheses to the power of 1 over x end exponent close square brackets
ln y equals limit as x rightwards arrow 0 of 1 over x ln open parentheses 1 plus fraction numerator 2 x over denominator 3 end fraction close parentheses
U sin g space L apostrophe H o s p i t a l apostrophe s space R u l e comma
ln y equals limit as x rightwards arrow 0 of open parentheses fraction numerator open parentheses begin display style fraction numerator 1 cross times 2 over denominator open parentheses begin display style fraction numerator 3 plus 2 x over denominator 3 end fraction end style close parentheses end fraction end style close parentheses over denominator 1 end fraction close parentheses
ln y equals limit as x rightwards arrow 0 of open parentheses fraction numerator 6 over denominator 3 plus 2 x end fraction close parentheses
ln y equals open parentheses fraction numerator 6 over denominator 3 plus 2 open parentheses 0 close parentheses end fraction close parentheses
ln y equals open parentheses 6 over 3 close parentheses
ln y equals 2
e to the power of ln y end exponent equals e squared
therefore y equals e squared

    L’Hospital’s rule is a general method of evaluating indeterminate forms such as 0/0 or ∞/∞. L’Hospital’s Rule tells us that if we have an indeterminate form 0/0 or /∞, all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.

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