Calculus

Limits and Continuity of Functions

Properties of Limits

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Problem 1
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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:
\[{\lim\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] }={ \lim\limits_{x \to a} f\left( x \right) + \lim\limits_{x \to a} g\left( x \right).}\]

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

\[\lim\limits_{x \to a} {a^{f\left( x \right)}} = {a^{\lim\limits_{x \to a} f\left( x \right)}},\] where the base \(a \gt 0\).

Limit of a Logarithm of a Function

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

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

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Problem 1
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Problems 2-5