Properties of Limits of Mathematical Functions in Calculus

Properties of limits of functions, in the form of theorems, are presented along with some examples of applications and detailed solutions.

Theorem: If f and g are two functions and both limx→a f(x) and limx→a g(x) exist, then
1. lim [ f(x) + g(x) ] = lim f(x) + lim g(x) : The limit of the sum of two functions is the sum of their limits.

    Example 1: Calculate limx→-2 h(x) where h(x) is given by
    h(x) = x + 5
    Solution to Example 1:
    We may consider h(x) as the sum of f(x) = x and g(x) = 5 and apply theorem 1 above
    limx→-2 h(x) = limx→-2 x + limx→-2 5
    x and 5 are basic functions and their limits are known.
    limx→-2 x = -2
    and
    limx→-2 5 = 5
    Hence, limx→-2 h(x) = -2 + 5 = 3

2. lim [ f(x) - g(x) ] = lim f(x) - lim g(x) : The limit of the difference of two functions is the difference of their limits.

    Example 2: Calculate limx→10 h(x) where h(x) is given by
    h(x) = x - 7
    Solution to Example 2:
    We may consider h(x) as the difference of f(x) = x and g(x) = 7 and apply theorem 2 above
    limx→10 h(x) = limx→10 x - limx→10 7
    x and 7 are basic functions with known limits.
    limx→10 x = 10
    and
    limx→10 7 = 7
    Hence, limx→10 h(x) = 10 - 7 = 3

3. lim [ f(x) * g(x) ] = lim f(x) * lim g(x) : The limit of the product of two functions is the product of their limits.
    Example 3: Calculate limx→-5 m(x) where m(x) is given by
    m(x) = 3 x
    Solution to Example 3:
    Let m(x) = f(x) * g(x), where f(x) = 3 and g(x) = x and apply theorem 3 above
    limx→-5 m(x) = limx→-5 3 * limx→-5 x
    3 is a constant function and x is also a basic function with known limits.
    limx→3 = 3
    and
    limx→- 5 x = - 5
    Hence,
    limx→-5 m(x) = 3*(- 5) = - 15

4. lim [ f(x) / g(x) ] = lim f(x) / lim g(x) ; if lim g(x) is not equal to zero. The limit of the quotient of two functions is the quotient of their limits if the limit in the denominator is not equal to 0.
    Example 4: Calculate limx→3 r(x) where r(x) is given by
    r(x) = (3 - x) / x
    Solution to Example 4:
    Let r(x) = f(x) / g(x), where f(x) = 3 - x and g(x) = x and apply theorem 4 above
    limx→3 r(x) = limx→3 (3 - x) / limx→3 x
    3 - x is the difference of two basic functions and x is also a basic function.
    limx→3 (3 - x) = 3 - 3 = 0
    and
    limx→3 x = 3
    Hence, limx→3 r(x) = 0 / 3 = 0
5. lim nth root [ f(x) ] = nth root [ lim f(x) ]. If n is even, lim f(x) has to be positive. The limit of the nth root of a function is the nth root of the limit of the function, if the nth root of the limit is a real number.
    Example 5: Calculate limx→5 m(x) where m(x) is given by
    m(x) = SQRT[2 x - 1]
    Solution to Example 5:
    Let f(x) = 2 x - 1 and find its limit applying the difference and product theorems above
    limx→5 f(x) = 2*5 - 1 = 9
    We now apply theorem 5 since the square root of 9 is a real number.
    limx→5 m(x) = SQRT(9) = 3
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