# Calculus

Research and create functions to explain the concepts of limit and continuity

1 . Create a function , f (x , and pick a point c such that the limit of f (x ) as x approaches c from the right and the limit of f (x ) as x approaches from the left are equal and the function is continuous . Show the values of the limits and explain why the function is continuous . The explanation should be intuitive as well as mathematical . INCLUDE a graph of the function

Solution

Let ‘s consider f (x x2 [banner_entry_middle]

1 . And let ‘s investigate its continuity at the point x 0 . It means that for function f (x ) there should exist the limit on the left , limit on the right , they should be equal to each other and to the value of the function at this point

In our particular case the given definition will be presented in a following way

or (x 0

Taking into account that f (x 0 1 we can state that function under study at the point x 0 is continuous

2 . Create a function , f (x , and pick a point c such that the limit of f (x ) as x approaches c from the right and the limit of f (x ) as x approaches from the left are equal , the function is defined at the point c but the function is not continuous at c . Show the values of the limits and explain why the function is not continuous . The explanation should be intuitive as well as mathematical . INCLUDE a graph of the function

Solution ? 1 (is not equal to the limit of f (x ) as x approaches 0 from the right and from the left ) and hence function is not continuous at the point x 0

3 . Create a function , f (x , and pick a point c such that the limit of f (x ) as x approaches c from the right and the limit of f (x ) as x approaches from the left are equal , the function is not defined at the point c and the function is not continuous at c . Show the values of the limits and explain why the function is not continuous . The explanation should be intuitive as well as mathematical . INCLUDE a graph of the function

Solution

Let ‘s consider function

And let ‘s investigate its continuity in the point x 0

is absolutely the same as that , carried out in the first task However , in this case f (x 0 ) is not defined and hence function is not continuous at the point x 0 (for the function f (x ) at the point x 0 to be continuous , it is necessary that limits of f (x ) as x approaches 0 from the right and from the left were equal to each other and were equal to the f (x 0

4 . Create a function , f (x , and pick a point c such that the limit of f (x ) as x approaches c from the right and… [banner_entry_footer]

**Author:** Essay Raptor