Calculus

Limits and Continuity of Functions

Limits and Continuity Logo

Properties of Limits

  • 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:

    \[{\lim\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] }={ \lim\limits_{x \to a} f\left( x \right) + \lim\limits_{x \to a} g\left( x \right).}\]

    Extended Sum Rule

    \[{\lim\limits_{x \to a} \left[ {{f_1}\left( x \right) + \ldots + {f_n}\left( x \right)} \right] }
    = {\lim\limits_{x \to a} {f_1}\left( x \right) + \ldots + \lim\limits_{x \to a} {f_n}\left( x \right).}
    \]

    Constant Function Rule

    The limit of a constant function is the constant:

    \[\lim\limits_{x \to a} C = C.\]

    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:

    \[{\lim\limits_{x \to a} kf\left( x \right) }={ k\lim\limits_{x \to a} f\left( x \right).}\]

    Product Rule

    This rule says that the limit of the product of two functions is the product of their limits (if they exist):

    \[{\lim\limits_{x \to a} \left[ {f\left( x \right)g\left( x \right)} \right] }={ \lim\limits_{x \to a} f\left( x \right) \cdot \lim\limits_{x \to a} g\left( x \right).}\]

    Extended Product Rule

    \[{\lim\limits_{x \to a} \left[ {{f_1}\left( x \right){f_2}\left( x \right) \cdots {f_n}\left( x \right)} \right] }
    = {\lim\limits_{x \to a} {f_1}\left( x \right) \cdot \lim\limits_{x \to a} {f_2}\left( x \right) \cdots}\kern0pt{ \lim\limits_{x \to a} {f_n}\left( x \right).}
    \]

    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

    \[{\lim\limits_{x \to a} {\left[ {f\left( x \right)} \right]^p} }={ {\left[ {\lim\limits_{x \to a} f\left( x \right)} \right]^p},}\]

    where the power \(p\) can be any real number. In particular,

    \[\lim\limits_{x \to a} \sqrt[\large p\normalsize]{{f\left( x \right)}} = \sqrt[\large p\normalsize]{{\lim\limits_{x \to a} f\left( x \right)}}.\]

    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

    \[\lim\limits_{x \to a} {b^{f\left( x \right)}} = {b^{\lim\limits_{x \to a} f\left( x \right)}},\]

    where the base \(b \gt 0.\)

    Limit of a Logarithm of a Function

    \[{\lim\limits_{x \to a} \left[ {\log _b f\left( x \right)} \right] }={ \log_b \left[ {\lim\limits_{x \to a} f\left( x \right)} \right],}\]

    where the base \(b \gt 0.\)

    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

    \[{\lim\limits_{x \to a} g\left( x \right) = \lim\limits_{x \to a} h\left( x \right)} = {L,}\]

    then

    \[\lim\limits_{x \to a} f\left( x \right) = L.\]

    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 a problem to see the 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.

    \[{\lim\limits_{x \to 10} \left( {2x\lg {x^3}} \right) }
    = {\lim\limits_{x \to 10} 2x \cdot \lim\limits_{x \to 10} \lg {x^3} }
    = {2\lim\limits_{x \to 10} x \cdot \lg \left( {\lim\limits_{x \to 10} {x^3}} \right) }
    = {2 \cdot 10 \cdot \lg 1000 = 20 \cdot 3 = 60. }
    \]

    Page 1
    Problem 1
    Page 2
    Problems 2-5