![]() ![]() This is simply because when you have an inequality, there is a vast range of solutions that can satisfy the equation, a range not limited to a line. ![]() However, if you notice, there's a shading on the left hand side of both graphs, and a dashed line on the #y > x# graph. This graph represents the equation #y >= x#Īs I mentioned before, inequality equations look very much like linear ones. If we were document this mathematically using an inequality, we'd get something like this: Your across-the-street rival Joe looks at your purchase and responds "tut tut, still a lot less than what I have," and walks away with a smirk. You buy 300 chickens that you're going to cook at your restaurant tonight for a party. Let me use a real life example to communicate this. Rather, inequalities deal with more nebulous greater than/less than comparisons. #(x-alpha)#, #(x-beta)#, #(x-gamma)# and #(x-delta)# take up either a positive or negative value,Īnd hence the polynomial (as it is a product of these linear binomials) will take positive or negative valueĪnd can easily check the intervals, where the inequality is satisfied, giving us the result.Īs an example, one may like to see solution to this question.Īn inequality is simply an equation where (as the name implies) you don't have an equal sign. In these intervals, we can find that each of these linear binomial i.e. (Note at these values, value of polynomial will be zero.)įor example, if they are already in increasing order, these are #(-oo,alpha)#, #(alpha,beta)#, #beta,gamma)#, #(gamma,delta)# and #(delta.oo)#. Note that numbers #alpha#, #beta#, #gamma# and #delta# divide real number in five intervals. ![]() It could also be less than or less than or equal or greater than or equal, but the process is not much effected. If x is allowed to decrease without bound, f(x) approaches 2.Sign chart is used to solve inequalities relating to polynomials, which can be factorized into linear binomials. ![]() If x is allowed to increase without bound, f(x) in the graph below approaches 2. If x is allowed to decrease without bound, f(x) take values within and has no limit again. If x is allowed to increase without bound, f(x) take values within and has no limit. The graph below shows a periodic function whose range is given by the interval. These are symbols used to indicate that the limit does not exist. Note that - ? and + ? are symbols and not numbers. We writeĪs x approaches - 2 from the right, f(x) gets larger and larger without bound and there is no limit. This graph shows that as x approaches - 2 from the left, f(x) gets smaller and smaller without bound and there is no limit. In this example, the limit when x approaches 0 is equal to f(0) = 1. Note that the left and right hand limits are equal and we can write Note that the left hand limit and f(1) = 2 are equal. Note that the left and right hand limits and f(1) = 3 are all different. The graph below shows that as x approaches 1 from the left, y = f(x) approaches 2 and this can be written asĪs x approaches 1 from the right, y = f(x) approaches 4 and this can be written as We consider values of x approaching 0 from the left (x 0). Let g(x) = sin x / x and compute g(x) as x takes values closer to 0. In fact we may talk about the limit of f(x) as x approaches a even when f(a) is undefined. NOTE: We are talking about the values that f(x) takes when x gets closer to 1 and not f(1). In both cases as x approaches 1, f(x) approaches 4. We first consider values of x approaching 1 from the left (x 1). Let f(x) = 2 x + 2 and compute f(x) as x takes values closer to 1. Numerical and graphical approaches are used to introduce to the concept of limits using examples. Introduction to Limits in Calculus Introduction to Limits in Calculus ![]()
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