Prove that \[\lim_{h\to 0}\,\frac{h^3}{h}=0.\]

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Lesson Parent: 
Function Limits VIII: Limits Involving Zero
Problem: 

Prove that \[\lim_{h\to 0}\,\frac{h^3}{h}=0.\]

Answer: 

It is true that \[\lim_{h\to 0}\,\frac{h^3}{h}=0.\]

SAMPLE SOLUTION

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Solution: 

Recall that in general when

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we need to show that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>.

For this particular problem, we need to show that for every <Sign in to see all the formulas> there exists a <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then <Sign in to see all the formulas>. The important thing to note is that <Sign in to see all the formulas> except when $h=0$. When $h=0$ then $h^3/h$ is undefined since we cannot divide by zero. However, our function limit proof states that we evaluate the function $h^3/h$ for values of $h$ in <Sign in to see all the formulas>. In particular, <Sign in to see all the formulas> means that we evaluate the function $h^3/h$ at values of <Sign in to see all the formulas>. Therefore, when <Sign in to see all the formulas> then <Sign in to see all the formulas>.

Let's choose <Sign in to see all the formulas>. Now when <Sign in to see all the formulas> then <Sign in to see all the formulas> as discussed above. Furthermore, the delta we chose makes it such that <Sign in to see all the formulas>. Now doing some basic algebra we know that

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Now since <Sign in to see all the formulas> and <Sign in to see all the formulas> we know that <Sign in to see all the formulas>.