Squeeze Theorem - Calculus

 Squeeze Theorem - Calculus

 

Education The squeeze theorem in Calculus, also known as the pinching theorem or the sandwich theorem deals with the limit of a function. Although, there are existing numerous ways; algebraic, graphical or a combination of both that can be used to find the limit of a function, there are exceptional cases where both techniques are not considered to be the most appropriate method. Such case would include finding the limit of a function defined as the cosine of the reciprocal of any real number, x as the value of x approaches to zero and some other equivalent functions. Algebraically if the value of x be replaced with zero in this function, it might be said that the limit of this function does not exist since the reciprocal of

 zero is not defined. In this circumstance, the squeeze theorem can be a vital tool and for many calculus applications involving proof-writing and other mathematical analysis. The squeeze theorem states that at certain x-value, an element in a close interval within the domain of a function, not necessarily in the interior of this interval, the limit of this function is between the limit of two other functions whose values are defined in the lower and upper bounds of the interval. In a more precise language, let I be the interval that contains the x-value, a which is the limit point and the three functions; f, g, and h be defined on the interval, I except perhaps at a. It must be correct that for every number, x in the interval not equal to a, the

 value of the function, f is between the value of the functions g and h, given that h is the lower bound of f and h is the upper bound of f. By taking the limit of both sides of the resulting inequality as described by the word “between”, the result will lead to the limit of the function f is as well between the limit of the functions, g and h. Nonetheless, if a is the endpoint of the interval I, then the limit of g is the left–hand limit of f and the limit of h is the right-hand limit of f. By applying the previously known concept of limit, this follows that the limit of the function f, is equal to the limit of g and h, which must be a real number, say L. For additional calculus help, visit integralcalc.com