lim x→∞ (x^2-3x+7) / (x^3+10x-4)

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`lim_(x-gtoo)(x^2-3x+7)/(x^3+10x-4)`

If you try to evaluate the limit straight away, you will get oo/oo which is indeterminate. We can remove this by dividing both numerator and denominator by x^3.

`lim_(x-gtoo)((x^2-3x+7)/x^3)/((x^3+10x-4)/x^3)`

`lim_(x-gtoo)(1/x-3/x^2+7/x^3)/(1+10/x^2-4/x^3)`

Now we can evaluate the limit.

`lim_(x-gtoo)(1/x-3/x^2+7/x^3)/(1+10/x^2-4/x^3) = 0/1 = 0`

Therefore,

`lim_(x-gtoo)(x^2-3x+7)/(x^3+10x-4) = 0`

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`lim_(x-gtoo)(x^2-3x+7)/(x^3+10x-4)`

If you try to evaluate the limit straight away, you will get oo/oo which is indeterminate. We can remove this by dividing both numerator and denominator by x^3.

`lim_(x-gtoo)((x^2-3x+7)/x^3)/((x^3+10x-4)/x^3)`

`lim_(x-gtoo)(1/x-3/x^2+7/x^3)/(1+10/x^2-4/x^3)`

Now we can evaluate the limit.

`lim_(x-gtoo)(1/x-3/x^2+7/x^3)/(1+10/x^2-4/x^3) = 0/1 = 0`

 

Therefore,

`lim_(x-gtoo)(x^2-3x+7)/(x^3+10x-4) = 0`

 

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