![]() How can the derivative tell us whether there is a maximum, minimum, or neither at a point? The following so-called First Derivative Test is a procedure for finding relative extrema of a continuous function based on critical points and analyzing behaviour around the critical points: Theorem 5.67. We can instead use information about the derivative \(f'(x)\) to decide since we have already had to compute the derivative to find the critical values, there is often relatively little extra work involved in this method. The method of Section 5.5.1 for deciding whether there is a relative maximum or minimum at a critical value is not always convenient. Subsection 5.7.1 The First Derivative Test and Intervals of Increase/Decrease ¶ In this section, we discuss how we can tell what the graph of a function looks like by performing simple tests on its derivatives. Implicit and Logarithmic Differentiation.Derivatives of Exponential & Logarithmic Functions.Derivative Rules for Trigonometric Functions.Limits at Infinity, Infinite Limits and Asymptotes.Symmetry, Transformations and Compositions.Open Educational Resources (OER) Support: Corrections and Suggestions.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. ![]() ![]() 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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