Calculus Limit Problem:
Evaluate the function at the given numbers (correct to six decimal places). Use the results to guess the value of the limit, or explain why it does not exist.
f(x)=cosx-1/sinx
x=1, .5, .4, .3, .2, .1, .05, .01
What should I do? A table of values??
Yes,do a table of values.
To evaluate the function at the given numbers and determine the limit, you can create a table of values. Here's how you can do it:
1. Start by plugging in the given numbers into the function f(x) = (cos(x) - 1) / sin(x). Calculate the value of the function for each number. We'll take x = 1 as an example:
f(1) = (cos(1) - 1) / sin(1)
Using a calculator or mathematical software, evaluate cos(1) and sin(1) and substitute them into the equation:
f(1) ≈ (0.540302 - 1) / 0.841471 ≈ -0.459698 / 0.841471 ≈ -0.546302
Repeat this process for each given number, filling out the table of values.
2. After computing the values of the function for all the given numbers, try to identify any patterns or trends in the function's values as x approaches a certain value. Pay attention to the trend as x gets closer to zero.
3. If the values in the table approach a specific value as x gets closer to a certain number (in this case, zero), then you can make a guess about the value of the limit. However, if the values do not approach any specific value or diverge, the limit does not exist.
Analyzing the values in the table is crucial for understanding the behavior of the function and making an educated guess about the limit.