Determine if limit exist inx^9+1/x^7+1?x go to -1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the limit of function `(x^9+1)/(x^7+1), ` hence you need to substitute -1 for x in function such that:

`lim_(x-gt-1) (x^9+1)/(x^7+1) = ((-1)^9 + 1)/((-1)^7 + 1)`

You should remember that a negative number raised to an odd power gives a negative result, hence `(-1)^9 = -1`  and `(-1)^7=-1` .

`lim_(x-gt-1) (x^9+1)/(x^7+1) = (-1 + 1)/(-1 + 1) = 0/0`

You may use several methods to find the limit. One method is to use the following formula:

`x^n + 1 = (x+1)(x^(n-1)-x^(n-2)+....(-1)(n-1))`

Substituting 9 for n yields:

`x^9 + 1 = (x+1)(x^(8)-x^(7)+....+1)`

`x^7 + 1 = (x+1)(x^(6)-x^(5)+....+1)`

Hence, `lim_(x-gt-1) (x^9+1)/(x^7+1) = lim_(x-gt-1) ((x+1)(x^(8)-x^(7)+....+1))/((x+1)(x^(6)-x^(5)+....+1))`

Reducing by `(x+1)`  yields:

`lim_(x-gt-1) (x^9+1)/(x^7+1) = lim_(x-gt-1) (x^(8)-x^(7)+....+1)/(x^(6)-x^(5)+....+1)`

Notice that `-(x)^7 = -(-1)^7 = 1,`  hence the negative of each odd power of -1 yields 1:

`lim_(x-gt-1) (x^(8)-x^(7)+....+1)/(x^(6)-x^(5)+....+1) = (1+1+...+1)/(1+1+...+1)`

The are 9 terms added in brackets to numerator and 7 terms to denominator such that:

`lim_(x-gt-1) (x^(8)-x^(7)+....+1)/(x^(6)-x^(5)+....+1) = (9*1)/(7*1)`

`lim_(x-gt-1) (x^(8)-x^(7)+....+1)/(x^(6)-x^(5)+....+1) = 9/7`

Hence, evaluating the limit to the function `lim_(x-gt-1)(x^9+1)/(x^7+1), ` yields `9/7` .

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