![]() ![]() It is helpful to look at a graph of the function. Hence we must investigate the limit using other techniques. Note that the product rule does not apply here because does not exist. If and when is near (except possibly at ) and both and, then. Further, does not exist because does not settle down to one specific finite value as approaches 0. We investigate the left and right-hand limits of the function at 0 visually. Limits of Piecewise Defined Functions via One-Sided Limitsĭefine the Heaviside function as follows: This fact follows from application of the limit laws which have been stated up to this point. We note that if is a polynomial or a rational function and is in the domain of, then. ![]() Limits of Polynomials and Rational Functions The limit of a positive integer root of a function is the root of the limit of the function: The limit of a positive integer power of a function is the power of the limit of the function: The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0): The limit of a product is the product of the limits: Note that the Difference Law follows from the Sum and Constant Multiple Laws. The limit of a difference is the difference of the limits: The limit of a constant times a function is the constant times the limit of the function: By the Sum Law, we have, and we know how to evaluate the two limits on the right hand side of the last equation using the two special limits we discussed above: and. The limit of a sum is the sum of the limits:Įxample: We evaluate the limit. Let be a constant and assume that and both exist. If we write out what the symbolism means, we have the evident assertion that as approaches (but is not equal to), approaches. Since is constantly equal to 5, its value does not change as nears 1 and the limit is equal to 5. To evaluate this limit, we must determine what value the constant function approaches as approaches (but is not equal to) 1. Evaluate limits involving piecewise defined functions.Įxample: Suppose that we consider. Apply the Squeeze Theorem to find limits of certain functions. Explain why certain limits do not exist by considering one-sided limits. Construct examples for which the limit laws do not apply. Evaluate limits using the limit laws when applicable. Limit laws, greatest integer function, Squeeze Theorem Limits and Derivatives: Calculating Limits Using the Limit Laws
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