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

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

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

Answer: 

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

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 $0/h=0$ except when $h=0$. When $h=0$ then $0/h$ is undefined since we cannot divide by zero. However, our function limit proof states that we evaluate the function $0/h$ 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 $0/h$ 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> and is always less than every <Sign in to see all the formulas>. So for every <Sign in to see all the formulas> we can pick any <Sign in to see all the formulas> such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>.