LIMITS AND CONTINUITY IN CALCULUS. LEARNING OBJECTIVES. • Know how to evaluate and graph piecewise-defined functions. • Know how to evaluate. Limits of Piecewise Functions: Learn how to calculate the limits of piecewise functions. Find the limits from graphs of piecewise functions using one-sided limits. Author: Betty Volkman Country: New Zealand Language: English Genre: Education Published: 11 November 2014 Pages: 425 PDF File Size: 1.83 Mb ePub File Size: 7.73 Mb ISBN: 590-3-19753-428-4 Downloads: 46053 Price: Free Uploader: Betty Volkman ## Limits of Piecewise-Defined Functions : How to Calculate, Examples, Practice Problems, pictures

And so if we wanna say what is the limit of f of x as x approaches four, well this is a good scenario here because from both limits of piecewise functions left and the right as we approach x equals four, we're approaching the same value, and we know, that in order for the two side limit to have a limit, you have to be approaching the same thing from the right and the left.

And we are, and so this is going to be equal to two. Now what's the limit as x approaches two of f of x? Well, as x approaches two, we are going to be completely in this scenario right over here. Now interesting things do happen at x equals one here, our denominator limits of piecewise functions to zero, but at x equals two, this part of the curve is gonna be continuous so we can just substitute the value, it's going to be two plus two, over two minus one, which is four over one, which is equal to four.

Let's do another example.

• Limits of Piecewise Functions Lessons by MATHguide
• Determining limits using algebraic properties of limits: direct substitution

So we have another piecewise function, and so let's pause our video and figure out these things. Alright, now let's do this together.

## Limits of piecewise functions (practice) | Khan Academy

So what's the limit as x approaches negative one from the right? So if we're approaching from the right, when we are greater than or equal to negative one, we are in this part of our piecewise function, and so we would say, this is going to approach, this is gonna be two, to the negative one power, which is equal to one half.

What about if we're approaching limits of piecewise functions the left? Well, if we're approaching from the left, we're in this scenario right over here, we're to the left of x equals negative one, and so this is going to be equal to the sine, 'cause we're in this case, for our piecewise function, of negative one plus one, which is the sine of zero, which is equal to zero.

Now what's the two-sided limit as x approaches negative one limits of piecewise functions g of x? Limit Notation Within the last section see Evaluating Limitswe performed two limits.

### Limits of piecewise functions

The first limit can be written like so. It is pronounced as 'the limit of f x as x approaches -1 from the left. Again, the plus sign that is hovering in the exponent position mean that we are being asked to approach the x-value -1 but this time from the right side of There are two cases that can happen when dealing with limits.

The Limit Exists Limits of piecewise functions some a-value, if DNE is shorthand for 'does not exist. So, we will examine a situation for case 2. The graph below is of g x.

## Limits of piecewise functions (video) | Khan Academy

We will use this graph of g x to determine the following limit. To determine this limit, we have limits of piecewise functions first consider both of these limits separately. First, we will consider this limit.

To calculate this limit, we have to approach the x-value -3 from the left side.

We have to observe limits of piecewise functions y-values and see where they are headed as we move to the right, closer to the x-value As we move closer and closer to the x-value -3, we can see that the y-values remain constant at