Calculus
Limits and Continuity of Functions
Properties of Limits
Problem 1
Problems 2-5
Notation of Limit
The limit of a function is designated by \(f\left( x \right) \to L\) as \(x \to a\) or using the limit notation: \(\lim\limits_{x \to a} f\left( x \right) = L\).
Below we assume that the limits of functions \(\lim\limits_{x \to a} f\left( x \right)\), \(\lim\limits_{x \to a} g\left( x \right)\), \(\lim\limits_{x \to a} {f_1}\left( x \right)\), \(\ldots\), \(\lim\limits_{x \to a} {f_n}\left( x \right)\) exist.
Sum Rule
This rule states that the limit of the sum of two functions is equal to the sum of their limits:
Extended Sum Rule
= {\lim\limits_{x \to a} {f_1}\left( x \right) + \ldots + \lim\limits_{x \to a} {f_n}\left( x \right).}
\]
Constant Function Rule
The limit of a constant function is the constant:
Constant Multiple Rule
The limit of a constant times a function is equal to the product of the constant and the limit of the function:
Product Rule
This rule says that the limit of the product of two functions is the product of their limits (if they exist):
Extended Product Rule
= {\lim\limits_{x \to a} {f_1}\left( x \right) \cdot \lim\limits_{x \to a} {f_2}\left( x \right) \cdots}\kern0pt{ \lim\limits_{x \to a} {f_n}\left( x \right).}
\]
Quotient Rule
The limit of quotient of two functions is the quotient of their limits, provided that the limit in the denominator function is not zero:
{\lim\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}} }={ \frac{{\lim\limits_{x \to a} f\left( x \right)}}{{\lim\limits_{x \to a} g\left( x \right)}},\;\;\;}\kern-0.3pt
{\text{if}\;\;\lim\limits_{x \to a} g\left( x \right) \ne 0.}
\]
Power Rule
where the power \(p\) can be any real number. In particular,
If \(f\left( x \right) = x^n\), then
{\lim\limits_{x \to a} {x^n} = {a^n},\;n = 0, \pm 1, \pm 2, \ldots \;\;\;}\kern-0.3pt
{\text{and}\;\;a \ne 0,\;\;\text{if}\;\;n \le 0.}
\]
This is a special case of the previous property.
Limit of an Exponential Function
where the base \(a \gt 0\).
Limit of a Logarithm of a Function
where the base \(a \gt 0\).
The Squeeze Theorem
Suppose that \(g\left( x \right) \le f\left( x \right) \le h\left( x \right)\) for all \(x\) close to \(a\), except perhaps for \(x = a\). If
then
The idea here is that the function \(f\left( x \right)\) is squeezed between two other functions having the same limit \(L.\)
Solved Problems
Click on problem description to see solution.
✓ Example 1
Find the limit \(\lim\limits_{x \to 10} \left( {2x\lg {x^3}} \right)\).
✓ Example 2
Find the limit \(\lim\limits_{x \to 9} {\large\frac{{4{x^2}}}{{1 + \sqrt x }}\normalsize}\).
✓ Example 3
Suppose that \(\lim\limits_{x \to 1} f\left( x \right) = 2\) and \(\lim\limits_{x \to 1} g\left( x \right) = 3.\) Calculate the limit \(\lim\limits_{x \to 1} {\large\frac{{g\left( x \right) – 3f\left( x \right)}}{{{f^2}\left( x \right) + g\left( x \right)}}\normalsize}.\)
✓ Example 4
Calculate the limit \(\lim\limits_{x \to \infty } {\large\frac{{3x + \cos x}}{{2x – 7}}\normalsize}.\)
✓ Example 5
Find the limit \(\lim\limits_{x \to \infty } {\large\frac{{2\sin x – 5x}}{{3x + 1}}\normalsize}\).
Example 1.
Find the limit \(\lim\limits_{x \to 10} \left( {2x\lg {x^3}} \right)\).
Solution.
= {\lim\limits_{x \to 10} 2x \cdot \lim\limits_{x \to 10} \lg {x^3} }
= {2\lim\limits_{x \to 10} x \cdot \lg \left( {\lim\limits_{x \to 10} {x^3}} \right) }
= {2 \cdot 10 \cdot \lg 1000 = 20 \cdot 3 = 60. }
\]
Problem 1
Problems 2-5