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Calculus 3 : Limits

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Example Question #351 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0+,0+)}-\frac{(2xy^{3} - 5x^{4} + 3y^{4})}{(x^{4} + 2y^{4})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }0\\&y=x:-24\end{align*}$

$\begin{align*}&\text{y-axis: }-\frac{3}{2}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:-\frac{3}{10}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{51}{2}\\&y=x:0\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }-\frac{3}{2}\\&y=x:0\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&-\frac{(2(0)y^{3} - 5(0)^{4} + 3y^{4})}{((0)^{4} + 2y^{4})}=\frac{-3y^{4}}{2y^{4}}\\&\lim_{(0,y)\rightarrow(0+,0+)}\frac{-3y^{4}}{2y^{4}}=-\frac{3}{2}\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0+,0+)}-\frac{(2x(x)^{3} - 5x^{4} + 3(x)^{4})}{(x^{4} + 2(x)^{4})}=0\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #352 : Limits

$\begin{align*}&\text{Calculate the limits of the function}\\&\lim_{(x,y)\rightarrow(0-,0+)}\frac{(26x^{3}y^{2} - 14x^{5})}{(18xy^{4} - 5x^{4}y + 3x^{5} + 3y^{5})}\\&\text{along the y-axis and the line, }y=x\end{align*}$

Possible Answers:

$\begin{align*}&\text{y-axis: }-\frac{120}{19}\\&y=x:0\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{12}{19}\end{align*}$

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{6}{133}\end{align*}$

$\begin{align*}&\text{y-axis: }\frac{36}{19}\\&y=x:0\end{align*}$

Correct answer:

$\begin{align*}&\text{y-axis: }0\\&y=x:\frac{12}{19}\end{align*}$

Explanation:

$\begin{align*}&\text{To approach this problem, it is important to understand what}\\ &\text{it is asking. Along the y-axis, }x=0\text{. Let us make that substitution:}\\&\frac{(26(0)^{3}y^{2} - 14(0)^{5})}{(18(0)y^{4} - 5(0)^{4}y + 3(0)^{5} + 3y^{5})}=\frac{0}{3y^{5}}\\&\lim_{(0,y)\rightarrow(0-,0+)}\frac{0}{3y^{5}}=0\\&\text{We can make a similar substitution for }y=x:\\&\lim_{(x,x)\rightarrow(0-,0-)}\frac{(26x^{3}(x)^{2} - 14x^{5})}{(18x(x)^{4} - 5x^{4}(x) + 3x^{5} + 3(x)^{5})}=\frac{12}{19}\\&\text{As these two limits are different, the limit does not exist.}\end{align*}$

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Example Question #351 : Limits

Evaluate:

$\lim_{(x,y)\rightarrow (0,1)} \frac{x^2+xy}{y}$

Possible Answers:

$\infty$

Does not exist

$-\infty$

Correct answer:

Explanation:

Since we won't have any divided by zero problems, we can just evaluate at the point.

$\lim_{(x,y)\rightarrow (0,1)} \frac{x^2+xy}{y}=\frac{0^2+0\cdot1}{1}=0$

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Example Question #352 : Limits

Evaluate:

$\lim_{(x,y)\rightarrow (0,1)} \frac{x^2+xy}{y}$

Possible Answers:

$-\infty$

Does not exist

$\infty$

Correct answer:

Explanation:

Since we won't have any divided by zero problems, we can just evaluate at the point.

$\lim_{(x,y)\rightarrow (0,1)} \frac{x^2+xy}{y}=\frac{0^2+0\cdot1}{1}=0$

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Calculus 3 : Limits

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #351 : Limits

Example Question #352 : Limits

Example Question #351 : Limits

Example Question #352 : Limits

All Calculus 3 Resources

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