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Limits and Continuity

Introduction to Limits

Limit is a concept that distinguishes calculus from other area of mathematics such as algebra and trigonometry. The concept of limit can be understood by considering examples such as finding the area of a region, slope of the tangent to a curve, the velocity of car and the sum of an infinite series.

The idea of limits give us a method for describing how the outputs of a function behave as the inputs approach some specified value. In this chapter, we show how to define and calculate limits of function values. The calculation rules are straightforward and most of the limits we need can be found by substitution, graphical investigation, numerical approximation, algebra or some combination of these.

Definition of limit of a function: A function f(x) is said to tend to a limit 'L' when 'x' tends to 'a', if the difference between f(x) and 'L' can be made as small as we please by making 'x' sufficiently near 'a' and we write: f(x) = L or f(x) = L.

The sentence f(x) = L or f(x) = L is read as: The limit of f(x) as 'x' approaches 'a' equals 'L'. The notation means that the values of the function f(x) approach 'L' as the values of 'x' approach 'a' (but do not equal 'a').

  • Ex: Guess the value of .

    Sol: Notice that the function f(x) = is not defined when x = 2, but that doesn't matter because the definition of f(x) says that we consider values of 'x' that are close to 'a' but not equal to 'a'.

    x < 2 f(x) x > 2 f(x)
    1.5 3.50000000 2.5 4.50000000
    1.9 3.90000000 2.1 4.10000000
    1.99 3.99000000 2.01 4.01000000
    1.999 3.99900000 2.001 4.00100000
    1.9999 3.99990000 2.0001 4.00010000
    1.99999 3.99999000 2.00001 4.00001000
    1.999999 3.99999900 2.000001 4.00000100
    1.9999999 3.99999990 2.0000001 4.00000010
    1.99999999 3.99999999 2.00000001 4.00000001
    The above table gives values of f(x) [corrected to eight decimal places] for values of 'x' that approach 2 [but are not equal to 2]. On the basis of the values in the table, we make the guess that: = 4.

Two simple observations:
(i) For a constant value 'c',   (c) = c.
(ii) For an identity function 'x',   (x) = a.

Properties of Limits

By applying following six basic properties about limits, we can calculate many unknown limits from limits we already know.

Six basic properties
If P, Q, a, k and n are real numbers and f(x) = P and g(x) = Q, then:
i) Sum rule: [f(x) + g(x)] = f(x) + g(x) = P + Q.
Therefore, the limit of the sum of two functions is the sum of their limits.

Ex: Find [x3 + x2 + x].
Sol: [x3 + x2 + x] = [x3] + [x2] + [x] = a3 + a2 + a.
ii) Difference rule: [f(x) – g(x)] = f(x) – g(x) = P – Q.
Therefore, the limit of the difference of two functions is the difference of their limits.

Ex: Find [x3 – x2 – x].
Sol: [x3 – x2 – x] = [x3] – [x2] – [x] = a3 – a2 – a.
iii) Product rule: [f(x) * g(x)] = f(x) * g(x) = P * Q.
Therefore, the limit of a product of two functions is the product of their limits.

Ex: Find [x2 (x + 2)].
Sol:
iv) Constant multiple rule: [k.f(x)] = k. f(x) = k.P.
Therefore, the limit of a constant times a function is the constant times the limit of the function.

Ex: Find [2x2 + 3x].
Sol: [2x2 + 3x] = [2x2] + [3x] = 2.[x2] + 3.[x] = 2a2 + 3a.
v) Quotient rule: .
Therefore, the limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not equal to zero.

Ex: Find .
Sol:
vi) Power rule: [f(x)]n = [ f(x)]n = Pn.
Therefore, the limit of a power of a function is that power of the limit of the function.

Ex: Find (x2 + x)2.
Sol: (x2 + x)2 = { (x2 + x)}2 = { (x2) + (x)}2 = {a2 + a}2.
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 .

    Sol:

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 .

    Sol:
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