8a38f4…: “From a fall equation”

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1 : Complete the following table using exact values.

$\begin {array}{|c|c|c|c|c|c|c|c|c|} \hline x & \frac {1}{3} & \frac {1}{30} & \frac {1}{300} & \frac {1}{301} & \frac {1}{1000} & \frac {1}{1001} & \frac {1}{50000} & \frac {1}{100004}\\\hline \frac {1}{x} & \answer {3} & \answer {30} & \answer {300} & \answer {301} & \answer {1000} & \answer {1001} & \answer {50000} & \answer {100004} \\\hline \end {array}$

1.1 : Based on the table above, make a conjecture about $\lim _{x\to 0^{+}}\frac {1}{x}$ . Does it exist? Explain.

It seems like the limit does not exist. The values $\frac {1}{x}$ don’t seem to approach any number, even when $x$ is very close to 0. It seems that these values just grow and grow and keep growing.

1.2 : Based only on the table of values, can you determine for certain if the limit exists or not?

Yes, the pattern is quite clear. Yes, because we have points that are close enough. No, because you can never be sure if you have tested points close enough. No, you can never know if a limit truly exists. No. Nothing exists. This is the matrix.

1.3 : Do you need to use a limit to determine the behavior of $f(x)$ at $x=0$ ?

No, but we already filled out the table, so why not? No, but it is an easy technique to use. Yes, limits are always needed to determine function’s behavior. Yes, because the function is not defined at $x=0$ so we need to use a limit.