Limit: Other Roots

Question

\lim _{x\to 5}\frac{\sqrt[3]{x+3}-2}{x-5}

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Use the substitution U = \sqrt[3]{x+3} which can also be written U^3 = x+3. Rewrite the original limit in terms of the new variable u. Notice, you need to replace both the radical in the numerator and the linear factor in the denominator. Also, when x approaches 5, u approaches some other value. The new limit is an algebraic limit that can be evaluated in the usual way

Answer 1

First, simplify the expression using substitution.

((x+3)^1/3-2)/(x-5)

let u = (x+3)^1/3 x = u^3-3 x-5 = u³-5-3 = u³-8 = (u-2)(u²+2u+4)

When x→5, u→2

lim u→2 (u-2)/(u³-8) = lim u→2 (u-2)/((u-2)(u²+2u+4)) = lim u→2 1/(u²+2u+4) = 1/(4+4+4) = 1/12