A couple numerical limits

Here are a couple of numerical limit problems, like the problems we discussed in our second day of class.

The problems

  1. Table 1 below shows some values of \(f(x)\) near \(x=3\). Use that table to give your best estimate of \(\lim_{x\to3} f(x)\). Be sure to provide a number of digits that you can justify and explain your justification.

    Table 1
    \(x\) \(f(x)\)
    3.1 1.74618943088019
    3.01 1.62245912673256
    3.001 1.61073375272935
    3.0001 1.60956743390228
    3.00001 1.60945086395969

  2. Let \(f(x) = (4^x-1)/x\). Compute the values of \(f(x)\) at \[x = 0.1,\, 0.01,\, 0.001,\, 0.0001, \text{ and } 0.00001\] and list those results in a table. Use your table to give your best estimate of \[\lim_{x\to0} \frac{4^x-1}{x}.\] Again, be sure to provide a number of digits that you can justify and explain your justification.