# 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.

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### 2 Answers

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.

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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