Properties of Limits of 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

Property 1: The limit of the sum of two functions is the sum of their limits.

lim [ f(x) + g(x) ] = lim f(x) + lim g(x)

Example 1

Calculate lim x→-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
lim x→-2 h(x) = lim x→-2 x + lim x→-2 5
x and 5 are basic functions and their limits are known.
lim x→-2 x = -2
and
lim x→-2 5 = 5
Hence, lim x→-2 h(x) = -2 + 5 = 3

Property 2: The limit of the difference of two functions is the difference of their limits.

lim [ f(x) - g(x) ] = lim f(x) - lim g(x) :

Example 2

Calculate lim x→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

Property 3: The limit of the product of two functions is the product of their limits

lim [ f(x) × g(x) ] = lim f(x) × lim g(x) :

    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

Property 4: 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.

lim [ f(x) / g(x) ] = lim f(x) / lim g(x) ; if lim g(x) is not equal to zero.

    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

Property 5: 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.

lim n √[ f(x) ] = n √[ lim f(x) ]. If n is even, lim f(x) has to be positive.

    Example 5

    Calculate limx→5 m(x) where m(x) is given by
    m(x) = √[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) = √(9) = 3

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