Evaluation of Algebraic Limits
Let f(x) be an algebraic function and let 'a' be any real number. Then, f(x) is known as an algebraic limit. We generally evaluate algebraic limits by using following five methods:
(i) Direct substitution method
(ii) Factorization method
(iii) Rationalization method
(iv) Method of evaluating limits of the form
(v) Method of evaluating limits when x ∞.
i) Direct substitution method: Directly substitute the value in the given expression and if we get a finite number, then that finite number is the limit of the given expression.
- Ex: A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after 't' minutes using the equation: n = 2t2 – 12t + 15. Find .
ii) Factorization method: Consider . If by substituting x = a, the rational function takes the form , ....., etc., then (x – a) is a factor of f(x) and g(x). In such a case, we factorize the numerator and denominator and then cancel out the common factor (x – a). After canceling out the common factor (x – a), we again substitute x = a in the given expression and see whether we get a meaningful number or not, this process is repeated till we get a meaningful number.
- Ex: A warm can of soft drink is placed in a cold refrigerator. The relationship between the temperature of the soft drink (T in degree Celsius) and the time (t in hours) is given as: T = . Find .
Sol: When t = 3 hours, the function T = assumes the form .
iii) Rationalization method: This method is particularly used when either numerator or denominator or both of these involve expressions consisting of square roots. In this method, we multiply the numerator and denominator of a rational function by the conjugate of either the numerator or the denominator.
iv) Method of evaluating limits of the form : In this method, we can evaluate the limits using the standard results: = n.an – 1 and am – n, where 'a' > 0 and 'm', 'n' Q (rational numbers set).
v) Method of evaluating limits when x ∞: Write down the given expression in the form of a rational function, that is, , if it is not so. If 'k' is the highest power of 'x' in numerator and denominator, then divide each term in numerator and denominator by xk. Finally, use the result when 'n' > 0.
- Ex: Find .