2.4 the precise definition of a limit

Question 1

How to prove the value of a limit using epsilon and delta

Answer 1

1. Write what you are going to prove. Eg given any epsilon, we can find delta such that In absolute notation, write x minus the number it is approaching and set that less than delta then the function given (in absolute value notation with all numbers to the left side) is less than epsilon
2. the function given (in absolute value notation with all numbers to the left side less than epsilon and simplify as much as you can
3. Your result will have epsilon over a number. So write chose delta to equal epsilon over the number or something smaller
4. Write proof colon then write give me epsilon. I choose delta equals epsilon over the number. Assume
   write x minus the number it is approaching and set less than epsilon over the number
5. Do the same steps as before but reverse with in absolute value notation x minus the number it is approaching less then episodio over a number. Isolate for epsilon. try and make the equation look like the function given (in absolute value notation with all numbers to the left side less than epsilon by doing the opposite of simplifying

Question 2

How to find delta given epsilon graph

Answer 2

1. Write find delta such that the absolute value of x minus the main dot x is less than delta coma the absolute value of g of x minus the main dot y set that less than the distance between the above and below number of that number
2. Find the greatest value of delta of the main dot x value and the numbers beside it. then write greatest delta equals that number

Question 3

Find delta given delta and epsilon (no graph given)

Answer 3

1. Draw the axes of the graph
2. For the equation less than a number, whatever x is subtracted by plot that on the y axes
3. For the equation less than delta, whatever the c is subtracted by put that on the x-axis
4. For the equation less than a number, that is the distance for the number above and below the number on the y axis
5. We need to find the number before and after the number on the x axis. So take out the x in the less than a number equation and turn that into and equation (f(x) equals). Substitute y for f(x) and isolate for y. Your answer should be what to do the y values to get the corresponding x value of that y value
6. Subtract those values for the main dot x value. Whichever answer is smaller will be your answer. But the answer in a box