Hi,please help me with this function f:R-> R,f(x)= x^2+ax+5/sqrt (x^2 +1). I must to calculate asymptote to -> infinity,if a=0.Thank you!!

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Supposing that you want to find the slant asymptote to `oo` , then you need to remember the form of the slant asymptote such that:

`y = mx + n`

`m = lim_(x-gtoo) (f(x))/x`

`n = lim_(x-gtoo) [f(x) - mx]`

You need to evaluate m putting a = 0 such that:

`lim_(x-gtoo) (x^2 + 5)/(xsqrt(x^2+1))`

You need to force `x^2`  factor both numerator and denominator such that:

`lim_(x-gtoo) (x^2(1 + 5/(x^2)))/(xsqrt(x^2(1+1/(x^2))))`

You need to remember that `sqrt(x^2) = |x|.`

You only need to select the positive value since `x-gt+oo` , hence `(xsqrt(x^2(1+1/(x^2)))) = (x^2sqrt(1+1/(x^2)))`

`m = lim_(x-gtoo) (x^2(1 + 5/(x^2)))/(x^2sqrt(1+1/(x^2)))`

Reducing `x^2`  yields:

`m = lim_(x-gtoo) (1 + 5/(x^2))/(sqrt(1+1/(x^2))) = 1`

You need to evaluate n such that:

`n = lim_(x-gtoo) (x^2 + 5)/(sqrt(x^2+1)) - x`

You need to bring the terms to a common denominator such that:

`n = (x^2 + 5 - xsqrt(x^2+1))/(sqrt(x^2+1))`

You need to force`x^2 ` factor under the square root such that:

`n = (x^2 + 5 - x^2sqrt(1+1/(x^2)))/(sqrt(x^2+1))`

You need to force `x^2`  factor both numerator and denominator such that:

`n = x^2(1+ 5/(x^2) - sqrt(1+1/(x^2)))/(xsqrt(1/(x^2)))`

`n = oo`

Hence, since n has no finite value, then there is no horizontal or slant asymptote to `oo.`

Sources:

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