# 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 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
lim
x→-2 h(x) = limx→-2 x + limx→-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, limx→-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 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

## 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