Determine the limit as 𝑡 tends to zero of nine 𝑒 to the power of six 𝑡 minus nine over 𝑡 by evaluating the function at the following values of 𝑡: plus or minus 0.5, plus or minus 0.1, plus or minus 0.01, plus or minus 0.001, and plus or minus 0.0001.
We have to evaluate this function of 𝑡 nine 𝑒 to the power of six 𝑡 minus nine over 𝑡 for various values of 𝑡. Let’s arrange these values in a table to make things easier for ourselves. Here’s our table. Notice that we’ve arranged the values of 𝑡 from a smallest negative 0.5 to largest 0.5. Now we need to evaluate our function for these values of 𝑡.
For example, in this cell, we need to put the value of nine 𝑒 to the power of 6 𝑡 minus nine over 𝑡, when 𝑡 is negative 0.5. This is something we can evaluate using a calculator. We get 17.104, correct to three decimal places. We evaluate the function when 𝑡 is negative 0.1 in much the same way. And we can continue to fill in our table using our calculator or better yet, a spreadsheet program.
Having filled in the table, you should get something like this. The question is, how do all these values help us determine the limit as 𝑡 tends to zero of nine 𝑒 to the power of six 𝑡 minus nine over 𝑡. This limit is the value that the function approaches as 𝑡 gets closer and closer to zero. We have two values of 𝑡 which are especially close to zero: negative 0.0001 and 0.0001. So we expect our limits to be very close to the values of the function for these values of 𝑡.
More than that, as 𝑡 gets closer and closer to zero from below and from above, we expect the value of the function to get closer and closer to this limit. So what value is very close to 53.984 and 54.016 and, more than that, is the value that the function is getting closer and closer to as 𝑡 gets smaller and smaller. The answer is 54.
This method involves a bit of guess work. And so this value is just an estimate. We could extend our table to include even smaller values of 𝑡. And we would see values of the function which were closer to 54 as a result. We can’t keep making 𝑡 smaller and smaller and smaller. At some point, we have to say what we think the limit is. At the moment, we can say that it looks very likely that this limit is 54. Later on, we’ll be able to prove this formally.