Problem to solve
Compute \[ \lim_{ x \to -\infty } \frac{3x^3+2}{\sqrt{x^4-2}} \]
Solution
STEP 1
Find the following limit: lim_(x->-∞) (3 x^3 + 2) (x^4 - 2)^(-1/2)
STEP 2
Hint: | Factor a square root out of the quotient. For large negative values of x: lim_(x->-∞) (3 x^3 + 2)/sqrt(x^4 - 2) = lim_(x->-∞)-sqrt((3 x^3 + 2)^2/(x^4 - 2)) = -lim_(x->-∞) sqrt((3 x^3 + 2)^2/(x^4 - 2)): -lim_(x->-∞)(sqrt((3 x^3 + 2)^2/(x^4 - 2)))
STEP 3
Hint: | Using the power rule, pass the limit through the 2nd root. lim_(x->-∞) sqrt((3 x^3 + 2)^2/(x^4 - 2)) = sqrt(lim_(x->-∞) (3 x^3 + 2)^2/(x^4 - 2)): sqrt(lim_(x->-∞) (3 x^3 + 2)^2/(x^4 - 2))
STEP 4
Hint: | The degree of the numerator of (3 x^3 + 2)^2/(x^4 - 2) is higher than that of its denominator, so it grows asymptotically faster as x approaches -∞. Since (3 x^3 + 2)^2 grows asymptotically faster than x^4 - 2 as x approaches -∞, lim_(x->-∞) (3 x^3 + 2)^2/(x^4 - 2) = ± ∞. Additionally (3 x^3 + 2)^2>0 and x^4 - 2>0 as x approaches -∞, so lim_(x->-∞) (3 x^3 + 2)^2/(x^4 - 2) = ∞: -sqrt(∞)
STEP 5
Hint: | Evaluate -sqrt(∞). -sqrt(∞) = -∞: Answer: | | -∞
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Edited by tododemate® - 2021