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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

**function.**

*continuous*** 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,