# 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 lim 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
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 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 lim _{x→10} h(x) = lim_{x→10} x - lim_{x→10} 7
x and 7 are basic functions with known limits. lim _{x→10} x = 10
and lim _{x→10} 7 = 7
Hence, lim _{x→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 lim_{x→-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 lim _{x→-5} m(x) = lim_{x→-5} 3 * lim_{x→-5} x
3 is a constant function and x is also a basic function with known limits. lim _{x→3 = 3
and
limx→- 5 x = - 5
Hence,
limx→-5 m(x) = 3*(- 5) = - 15
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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 lim_{x→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 lim _{x→3} r(x) = lim_{x→3} (3 - x) / lim_{x→3} x
3 - x is the difference of two basic functions and x is also a basic function. lim _{x→3} (3 - x) = 3 - 3 = 0
and lim _{x→3} x = 3
Hence, lim _{x→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 lim_{x→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 lim _{x→5} f(x) = 2*5 - 1 = 9
We now apply theorem 5 since the square root of 9 is a real number. lim _{x→5} m(x) = SQRT(9) = 3
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