Find the limit. (If you need to use or –, enter INFINITY or –INFINITY, respectively.)? lim x→− Infinity (−2x + 8) / √(x2 + x)
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You need to evaluate the following limit such that:
`lim_(x->-oo) (-2x + 8) /(sqrt(x^2+x)) = oo/oo`
You may force factor x to numerator such that:
`lim_(x->-oo) x(-2 + 8/x)/(sqrt(x^2+x))`
You may force factor `x^2` to denominator such that:
`lim_(x->-oo) x(-2 + 8/x)/(sqrt(x^2(1+x/x^2)))`
You should remember that `sqrt(x^2) = |x|` and since `x->-oo ` `=> |x| = -x` such that:
`lim_(x->-oo) x(-2 + 8/x)/(-xsqrt(1+1/x))`
Reducing x yields:
`lim_(x->-oo) (-2 + 8/x)/(-sqrt(1+1/x)) = (-2 + lim_(x->-oo) (8/x))/(-sqrt(1 + lim_(x->-oo) (1/x)))`
`lim_(x->-oo) (-2 + 8/x)/(-sqrt(1+1/x)) = (-2 + 0)/(-sqrt(1+0))`
`lim_(x->-oo) (-2 + 8/x)/(-sqrt(1+1/x)) = (-2)/(-1) = 2`
Hence, evaluating the given limit yields `lim_(x->-oo) (-2x + 8) /(sqrt(x^2+x)) = 2.`
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