Lesson 22 of 21
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# Lesson 2: Understanding Limits

By definition a limit is: A function f  has a limit L as x approaches a, written provided that the values of f(x) get closer and closer to L as x gets closer and closer to a, on both sides of a, but x ≠ a

Another way to state this definition is to say that a limit is the intended height (where height f is a function in term of time x) of a function

Determining the height of a function

Example 1:  This function changes height all throughout its domain.  That different “x” values have different heights.

What is the height of the above function when x = 2?

Solution: The function reaches a height of 4 when x = 2.  A limit statement can be made. which simply means that the function as x approaches 2 is getting closer to the height of 4.

Functions that don’t reach their intended heights

Functions that don’t reach their intended heights

Example 2: Can a function have a limit if it can’t reach its intended height?  This function appears to be a regular function; however, there is a hole in the middle of the line representing this function.

When x =2 the function is undefined.

When we substitute x = 2 into the function the following happens.  When Does a Limit Exist?

A limit exists if you travel along a function from the left side and from the right side towards some specific value of x. In other words, as long as the y-values from the left and right are the same then the limit exists.

Example 3: What is the Limit of the function shown below? Solution: For a limit to exist the left hand limit must equal the right hand limit. So, in this example the limit does not exist.

This is also called a discontinued function.  If the limit did exist it would be called a continuous function.

Discontinued Function: Continuous Function: Evaluating Limits (substitution) Example 4: Solution:

Substitute 3 in for n. Therefore, Note: If you substitute and there is an answer, the limit is the answer.

Example 5:

Solution:

Factor the numerator first then since a substitution yielded an undefined answer, cancel out common factors to the numerator and denominator and substitute. Therefore, Example 6: Find Solution: Conjugates are used to evaluate this question because we must rationalize the numerator (see Lesson 1).   Therefore, 