Continuous functions And limits

POP. If you have to lift your pencil to make the graph then its not continuous.

Continuous Examples

What is a limit? As x approaches from left X= X X= X As x approaches from right

Can Be Discontinuous and have a limit

Discontinuous with one-sided limits

How to find the limit of certain functions The following slides will help you find the limit of some basic polynomial and rational functions. This will not help you find more complicated functions. Worst case scenario take values really close to the limit and see if there is a trend.

Ways to find a limit Is the limit going to a constant? Ex. Is the limit going to infinity? Ex. YES

Is the function a polynomial function? Is the function a rational function with two polynomial functions? Limit as x goes to a constant YES

Is the function a polynomial function? Is the function a rational function with two polynomial functions? Limit as x goes to ∞ YES

Then just plug in the value of x. Polynomial functions are continuous for all real numbers. Return

Plug in the value of x Do you get ? Do you get ? YES

Then the limit goes to zero. EndReturn

Look at the highest degree of each polynomial. Is the highest degree in the numerator? Is the highest degree in the denominator? Are the degrees the same in both denominator and numerator? None of the above? YES

Then the limit goes to infinity Return

Then the limit is the fraction of the leading coefficients Return

Then the limit goes to zero The numerator grows slower than the denominator. It acts like 1/∞ Return

The function needs to be factored, or simplified, or multiply by the conjugate of the denominator. Then plug the value of x back in. You may also take values of x that are approaching what the limit is approaching. endReturn

The end In general finding a limit of a function is finding the value of f(x) as x approaches a value. In general finding a limit of a function is finding the value of f(x) as x approaches a value. The left side and the right side limits must be the same in order for it to have a have limit at the point. The left side and the right side limits must be the same in order for it to have a have limit at the point. The function does not have be continuous at the limit value. The function does not have be continuous at the limit value. A graph of the function might be helpful to find the limit, but you cant always depend on them. A graph of the function might be helpful to find the limit, but you cant always depend on them. WebsiteReturn