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Sign chart calculus limits9/14/2023 ![]() ![]() Again we look at the signs on top to determine what we are looking for. Now this line chart is the second derivative and this tells us the concavity. If there is a “nd” over the number then that means the point is not defined and if there is a “dc” above the numbers then there is a discontinuity on that x value. The reason there is a zero above the zero is because it shows us that there is a change in direction without there being a discontinuity and the point is defined. ![]() After zero since there are negative marks we know that the graph is decreasing. Since there are plus marks over the points leading to zero we know that the original graph is increasing at this time. We see that the chart shows us whether the equation is positive or negative at the points 3. Since this is the derivative line chart this tells us whether the original graph is increasing or decreasing. So I am going to show some sign charts and explain exactly what they mean. They are used to show critical points given by the derivative and second derivative of certain function. They can tell you when a graph is increasing or decreasing, if the graph has a discontinuity, or even the concavity of the original graph. Sign charts are almost like number lines. On a semi-related note, just because the left-hand and right-hand limits are equal as they approach some value of x, it does not mean the function is continuous at this point.So we have talked about concavity and whether a graph is increasing or decreasing, but in order for my next post to make sense we must go over what sign charts are. Otherwise, an ordinary does not exist, as seen above. Remember it is important to do this from both sides, so you must evaluate x = -0.9, x = -0.99, x = -0.999, to make sure the limit is the same as you approach x from both sides. By doing this you will quickly see that the function approaches some value, which is your limit. For example, if you want the limit as x approaches 1 but evaluating x = 1 is impossible. To fix this issue, you should sub values close to x, slowly getting closer and closer to x, and evaluate your function from both sides. ![]() Sometimes this may not be possible, as it may end up with the division of 0 for example. All you have to do is substitute the x value that you want the limit for, into your function. An easy method of finding a limit, if it exists, is the substitution method. Looking at your graph it easy to find the answer, which you have correctly said is 2.įinding a limit generally means finding what value y is for a value of x. The limit does not exist as x approaches 0.įinally, this is asking for the value of the function at x = 2. As the limits differ depending on direction, the answer should be the same as the question above. Checking your graph, we can easily see the limit as x approaches 0 from the right is -1. However, we must also check to see if the right-hand limit is the same. Using the same logic as above, we can see that the left-hand limit of the function as x approaches 0 is equal to 3. ![]() It is important to test the function from both sides of the limit. Thus, we can see that there is no limit as x approaches 2. However, as we see in the above answers, the limit as x approaches 2 is different depending on the direction. The third is asking for the limit as x approaches 2. Following the same logic but from the other direction, we again find your answer to be correct. The second asks for the right-hand limit (indicated by the plus sign) as x approaches 2. Doing this, you can clearly see you answer is correct. To find this you follow the graph of your function from the left of the curve to the right as x approaches 2. The first one is asking for the left-hand limit (indicated by the minus sign). Answering your questions from top to bottom: ![]()
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