Limits on the AP Calculus Exam A Review

Limits are one of the first topics we need to master in our calculus courses.   There are many times when a function does not take on a value at a specific point.   For instance, let’s say we take the function f(x) = sin(x)/x.   What is the value of this function at x = 0?   Clearly, we cannot divide by 0, so there is a hole in this graph—there is no value for the function at x = 0.   However, if we look at the graph below, we see that the function gets closer and closer to 1 as x gets closer and closer to 0.    This is the idea of a limit.   Even if we do not have the exact value of the function at a certain value of x, we might still know what value the function gets closer and closer to as we get closer and closer to a specific x. In other words, a limit is the value that a function approaches as an input approaches a specific value. AP Calculus Exam Review: Notation of Limits The AP exam writes limits in the following manner: The first example above is read as the ‘the limit of f(x), as x approaches n’.   If we look at the three examples, we’ll notice that sometimes you’ll see a + or a – in the limit.   If we see a +, this means from the right or from the positive.   A negative means from the left or from the negative.   We will discuss this in more depth below. How to Calculate Limits There are several scenarios that the AP Exam might ask about in regards to determining limits.   If the question exists on the calculator section, a graphing calculator can be used to quickly solve limits.   Simply graph the function, and set the window to the domain in which we are looking, and see what the graph is doing. For example, let us take the graph below: This is the graph of f(x) = 1/x.   There is no value for this function as x=0 (remember, we can never divide by 0 in any equation).   However, the graph shows that as we approach x= 0 from the right, the graph approaches positive infinity.   From the left, negative infinity.   The trace function on many graphing calculators is useful in figuring out the value a graph is approaching. However, most limit questions will not involve the calculator, or can be more quickly solved analytically.    3 Most Common Limit Scenarios There are a few scenarios that the AP exam might ask.   Almost all limit questions will involve a fraction: Scenario #1 In the first scenario, we  will be asked to evaluate a limit where the denominator evaluates to 0, but the numerator evaluates to a non-zero number.   In such a case, the answer is always infinity or negative infinity (can you see why this is the case?). To determine which, carefully look at the numerator and the denominator separately.   The numerator will always be positive or negative.   The denominator will be 0, but look at which way we are approaching from.   If from the positive, pick a number just to the right of the limit (in this case, we can take 3.1).   If it asked from the negative, we could take a number just to the left.   Does this now evaluate to positive or negative?   3.1-3 is positive, so we have a positive numerator and denominator, and our limit evaluates to positive infinity.   If our numerator and denominator are the same sign, our limit is positive infinity; Different signs, negative infinity. Scenario #2 The second scenario is if there is no fraction, or the denominator does not evaluate to 0.   Chances are, in this case, the exam is asking us to see what happens as the limit approaches infinity. In such a case, we just need to ask ourselves is the limit getting bigger and bigger forever, or is it approaching a certain number?   In the first example above, as x gets bigger, the function gets bigger, so the limit approaches infinity.   In the second, the function gets closer and closer to 2.   If we’re ever not sure, just plug in a very large number and try to get a sense of what the function is doing. Scenario #3 The third scenario involves limits where both the numerator and denominator both approach 0 or infinity. In the above example, if we plug in 5 to the function, we get 0/0, which doesn’t help us.   There are two common ways of solving these types of questions. First, we can sometimes factor the equations.   In the above example we get: We can see that the x-5 term in each equation cancels out, leaving us with (x+3)/(x+2).   Plug  in 5, and we get (5+3)/(5+2) = 8/7, the solution to our question. This won’t always be the case, and there is a great shortcut anytime we have a limit that evaluates to 0/0 or  ±infinity/ ±infinity: L’Hospital’s Rule.   If we take the derivative of the numerator and denominator separately, we can evaluate the limit. In our above example: Evaluating, we get (10-2)/(10-3) or 8/7, the same answer we got from factoring.   This is often a quicker way of solving than factoring.   In addition, it works on equations where factoring does not help.   For instance: We will have more posts soon with tricks and practice problems regarding limits.   Limits are a great place to start studying as they are an important foundation for much of your calculus course.

Aptitude Tests and Their Different Definitions

Aptitude tests are designed to find out if a specific person is able to do a job or is fit for a certain position or not. They have been the bane of existence for students for years. Worrying about your SATs and having nightmares about them is a rite of passage for every student who passes through the education system. As the name suggests, these tests are required to find the aptitude and abilities of a person. But in reality, all they do is cause hysteria and mass confusion. The most unfortunate thing about them is when you graduate at last and think that they have left your side, BOOM! There they are again to haunt you. Educational institutions are not the only places where you will have to pass these tests, but now many job interviews organize aptitude tests to narrow down the pool of applicants, like it wasn’t hard enough already to find a job in today’s competitive market. If you know the real meaning behind this accursed word, well and good. But if you don’t, you are one of the lucky few who are still living in the land where unicorns are real. Anyhow, in your case ignorance is bliss.   If you don’t really know the real meaning behind the term aptitude test, you can easily confuse it with something else like: 1. A Test That Tells If You Are Smart or Dumb As if patronizing teachers and the mounting heaps of assignments and term papers weren’t enough to do that already. 2.  An Exam Which In Case Of Failure Doesn’t Let You Enter the College Who needs college education anyway? It’s not like you are going to get a job after you graduate, so you might as well start thinking about a career in retail. 3.  A Set of Silly Questions Designed To Make You Feel Stupid Like knowing about the American Revolution and the Pythagoras theorem is going to prepare you for important things in life like filing taxes and qualifying for a mortgage. 4.  A Test That Checks Your Knowledge of a Code Language An aptitude test not only tests your patience but also makes you question the number of brain cells that you have. It might be about an ancient language. It’s time to finally apply your knowledge of Klingon to ace a test! 5.  A Test That Determines Your Suitable Boyfriend/Girlfriend Is it a sophisticated game of â€Å"Love me, Love me not?† You looking at your crush like Might this be the answer to all your relationship problems? If you are not familiar with the true purpose of aptitude tests, this can be a good possibility. 6.  A Test of Your Cell Phone Apps Knowledge Thank god! A test that you can actually pass! Because you always find time to check out the trending cell phone apps, and to update the existing ones on your cell phone. Your evening basically looks like If you are unfamiliar with the true purpose of an aptitude test, you might think it has something to do with the abovementioned things. Because anything is better than thinking that it can define your future and that your success or failure depends on it.