Problem to solve
Compute \[ \lim_{ x \to \infty } \frac{7x-4}{\sqrt{x^3+5}} \]
Solution
STEP 1
Find the following limit: lim_(x->∞) (7 x - 4) (x^3 + 5)^(-1/2)
STEP 2
Hint: | Factor a square root out of the quotient. (7 x - 4)/sqrt(x^3 + 5) = sqrt((7 x - 4)^2/(x^3 + 5)): lim_(x->∞)(sqrt((7 x - 4)^2/(x^3 + 5)))
STEP 3
Hint: | Using the power rule, pass the limit through the 2nd root. lim_(x->∞) sqrt((7 x - 4)^2/(x^3 + 5)) = sqrt(lim_(x->∞) (7 x - 4)^2/(x^3 + 5)): sqrt(lim_(x->∞) (7 x - 4)^2/(x^3 + 5))
STEP 4
Hint: | The degree of the numerator of (7 x - 4)^2/(x^3 + 5) is lower than that of its denominator, so it grows asymptotically slower as x approaches ∞. Since (7 x - 4)^2 grows asymptotically slower than x^3 + 5 as x approaches ∞, lim_(x->∞) (7 x - 4)^2/(x^3 + 5) = 0: sqrt(0)
STEP 5
Hint: | Evaluate sqrt(0). sqrt(0) = 0: Answer: | | 0
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Edited by tododemate® - 2021