Properties of limits of functions, in the form of theorems, are presented along with some examples of applications and detailed solutions.
In what follows, \( C \) represents the constant of integration where applicable.
Theorems & Properties of Limits
Theorem: If \( f \) and \( g \) are two functions and both \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist, then:
Property 1: The limit of the sum of two functions is the sum of their limits.
\[ \lim_{x \to a} [ f(x) + g(x) ] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \]Property 2: The limit of the difference of two functions is the difference of their limits.
\[ \lim_{x \to a} [ f(x) - g(x) ] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \]Property 3: The limit of the product of two functions is the product of their limits.
\[ \lim_{x \to a} [ f(x) \times g(x) ] = \lim_{x \to a} f(x) \times \lim_{x \to a} g(x) \]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.
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \quad \text{provided that} \quad \lim_{x \to a} g(x) \neq 0 \]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.
\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} \]If \( n \) is even, \( \lim_{x \to a} f(x) \) has to be positive.
Examples with Detailed Solutions
Click on each example to view its detailed step-by-step solution.
Example 1
Calculate \( \lim_{x \to -2} h(x) \) where \( h(x) \) is given by:
\[ h(x) = x + 5 \]Show 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:
\[ \lim_{x \to -2} h(x) = \lim_{x \to -2} x + \lim_{x \to -2} 5 \]\( x \) and \( 5 \) are basic functions and their limits are known:
\[ \lim_{x \to -2} x = -2 \quad \text{and} \quad \lim_{x \to -2} 5 = 5 \]Hence:
\[ \lim_{x \to -2} h(x) = -2 + 5 = 3 \]Example 2
Calculate \( \lim_{x \to 10} h(x) \) where \( h(x) \) is given by:
\[ h(x) = x - 7 \]Show 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:
\[ \lim_{x \to 10} h(x) = \lim_{x \to 10} x - \lim_{x \to 10} 7 \]\( x \) and \( 7 \) are basic functions with known limits:
\[ \lim_{x \to 10} x = 10 \quad \text{and} \quad \lim_{x \to 10} 7 = 7 \]Hence:
\[ \lim_{x \to 10} h(x) = 10 - 7 = 3 \]Example 3
Calculate \( \lim_{x \to -5} m(x) \) where \( m(x) \) is given by:
\[ m(x) = 3x \]Show Solution to Example 3
Let \( m(x) = f(x) \times g(x) \), where \( f(x) = 3 \) and \( g(x) = x \), and apply Theorem 3 above:
\[ \lim_{x \to -5} m(x) = \lim_{x \to -5} 3 \times \lim_{x \to -5} x \]3 is a constant function and \( x \) is a basic function with known limits:
\[ \lim_{x \to -5} 3 = 3 \quad \text{and} \quad \lim_{x \to -5} x = -5 \]Hence:
\[ \lim_{x \to -5} m(x) = 3 \times (-5) = -15 \]Example 4
Calculate \( \lim_{x \to 3} r(x) \) where \( r(x) \) is given by:
\[ r(x) = \dfrac{3 - x}{x} \]Show Solution to Example 4
Let \( r(x) = \dfrac{f(x)}{g(x)} \), where \( f(x) = 3 - x \) and \( g(x) = x \), and apply Theorem 4 above:
\[ \lim_{x \to 3} r(x) = \dfrac{\lim_{x \to 3} (3 - x)}{\lim_{x \to 3} x} \]\( 3 - x \) is the difference of two basic functions and \( x \) is a basic function:
\[ \lim_{x \to 3} (3 - x) = 3 - 3 = 0 \quad \text{and} \quad \lim_{x \to 3} x = 3 \]Hence:
\[ \lim_{x \to 3} r(x) = \dfrac{0}{3} = 0 \]Example 5
Calculate \( \lim_{x \to 5} m(x) \) where \( m(x) \) is given by:
\[ m(x) = \sqrt{2x - 1} \]Show Solution to Example 5
Let \( f(x) = 2x - 1 \) and find its limit applying the difference and product theorems above:
\[ \lim_{x \to 5} f(x) = 2 \times 5 - 1 = 9 \]We now apply Theorem 5 since the square root of 9 is a real number:
\[ \lim_{x \to 5} m(x) = \sqrt{9} = 3 \]