Find limits for lim x-->1 ((x^5)-1)/(x-1) Use L'Hospital's Rule when appropriate. If there is a more elementary algebraic method, consider it.Find limits for  lim x-->1  ((x^5)-1)/(x-1) Use...

Find limits for lim x-->1 ((x^5)-1)/(x-1) Use L'Hospital's Rule when appropriate. If there is a more elementary algebraic method, consider it.

Find limits for  lim x-->1  ((x^5)-1)/(x-1)

Use L'Hospital's Rule when appropriate. If there is a more elementary algebraic method, you might consider it.

If you are able to use L' Hopital's Rule and when you convert your expression to a fraction, check and state that the hypotheses of the theorem are satisfied. check problem by graphing the function near the limit and see if the values support your solution.

Asked on by cspanutius

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lfryerda's profile pic

lfryerda | High School Teacher | (Level 2) Educator

Posted on

To find this limit, we can factor the polynomial in the numerator, and cancel out any common factors.

`lim_{x->1} {x^5-1}/{x-1}`

`=lim_{x->1}{(x-1)(x^4+x^3+x^2+x+1)}/{x-1}`

`=lim_{x->1}(x^4+x^3+x^2+x+1)`

`=1+1+1+1+1` 

`=5`

where in the last two lines the denominator is no longer a problem with the limit and we can use direct substitution.

 

Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

lim x-->1  ((x^5)-1)/(x-1)

`lim_(x->1) (x^5 - 1)/(x-1)`

If we substitute in 1 for x, we end up with the indeterminate form 0/0. Therefore, we can use L'Hopital's Rule to get: 

`lim_(x->1) (5x^4)/(1)`

``which equals 5. 

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