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Chapter 9: Problem 722

\(\lim _{\mathrm{y} \rightarrow(0)+}\left[\left({ }^{3} \sqrt{\mathrm{y}}+{ }^{3} \sqrt{\mathrm{y}}^{2}-{ }^{4} \sqrt{\mathrm{y}}^{3}\right) /\left({ }^{3} \sqrt{\mathrm{y}}+\sqrt{\mathrm{y}}+{ }^{4} \sqrt{\mathrm{y}}^{3}\right)\right]\) (a) \(-1\) (b) Limit does not exist (c) 1 (d) 0

Short Answer

Expert verified
The short answer is: (b) Limit does not exist.

Step by step solution

01

Analyze the given function

The given function is \[ \lim _{y \rightarrow(0)+}\left[\frac{{ }^{3}\sqrt{y}+{ }^{3}\sqrt{y^{2}}-{ }^{4}\sqrt{y^{3}}}{{ }^{3}\sqrt{y}+\sqrt{y}+{ }^{4}\sqrt{y^{3}}}\right] \]
02

Simplify the expressions inside the limit function

Simplify the expressions inside the limit function as follows: \[ { }^{3}\sqrt{y^{2}} = { }^{3}\sqrt{(y^{\frac{1}{3}})^{2}} = (y^{\frac{1}{3}})^{2} \text{ and } { }^{4}\sqrt{y^{3}} = { }^{4}\sqrt{(y^{\frac{1}{4}})^{3}} = (y^{\frac{1}{4}})^{3} \] Now, we have: \[ \lim _{y \rightarrow(0)+}\left[\frac{{ }^{3}\sqrt{y}+(y^{\frac{1}{3}})^{2}-(y^{\frac{1}{4}})^{3}}{{ }^{3}\sqrt{y}+\sqrt{y}+(y^{\frac{1}{4}})^{3}}\right] \]
03

Factor the numerator and denominator

Factor the numerator and denominator by factoring out the common terms y^{\frac{1}{3}} and y^{\frac{1}{4}} respectively: \[ \lim _{y \rightarrow(0)+}\left[\frac{y^{\frac{1}{3}}(1+y^{\frac{1}{3}}-y^{\frac{1}{2}})}{y^{\frac{1}{4}}({ }^{3}\sqrt{y}+\sqrt{y}+(y^{\frac{1}{4}})^{2})}\right] \]
04

Simplify the limit function by canceling out the common factors

We can cancel out the common factors y^{\frac{1}{3}} and y^{\frac{1}{4}} from the numerator and denominator: \[ \lim _{y \rightarrow(0)+}\left[\frac{1+y^{\frac{1}{3}}-y^{\frac{1}{2}}}{(y^{\frac{1}{4}})({ }^{3}\sqrt{y}+\sqrt{y}+(y^{\frac{1}{4}})^{2})}\right] \]
05

Substitute the limit into the simplified function and find the limit

Now, substitute the limit (y approaching 0) into the simplified function: \[ \lim _{y \rightarrow(0)+}\left[\frac{1+0^{\frac{1}{3}}-0^{\frac{1}{2}}}{(0^{\frac{1}{4}})(0^{\frac{1}{3}}+0+0^{\frac{1}{2}})}\right] \] Evaluating the expression, we have: \[ \frac{1+0-0}{(0)(0+0+0)} \] Since we have a 0 in the denominator, the limit does not exist. So, the answer is (b) Limit does not exist.

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