Exponential and Logarithmic Limits: One of the most important functions in Mathematics is the exponential function. The logarithmic function, the inverse of exponential functions, has a wide range of applications. These functions are used in a variety of real-world situations. Limits explain the behaviour of a function near a point rather than at that point.

The foundation of calculus is this basic yet strong principle of limits. The limits of the exponential and logarithmic functions can be easily studied from their graphs. These limits helps in many real world applications to come to a resolution.

What are Exponential Functions?

The term exponent means the power of a number. For example, the exponent of \(7\) in \(7^{3}\) is \(3\) . Similarly, the exponential functions are those where the variable occurs as a power. An exponential function is defined as:

\(f(x)=a^{x}\)

where \(a\) is a positive real number, not equal to \(1\) .

The most used exponential function base is the transcendental number \(e\). The value of \(e\) is \(2.71828\). Using the base as \(e\) we can represent the exponential function as

\(y=e^{x}\)

It is called a natural exponential function.

If \(a=1\), then \(f(x)=1^{x}\), which is equal to \(1, \forall x\). Hence the graph of the function would just be a straight line of constant \(y(=1)\).

Cases of Exponential Functions

Depending on the value of ‘ \(a\) ‘, we can have two possible cases of exponential functions:

Case 1: \(a>1\) The exponential function increases rapidly with increasing \(x\). The function tends to \(+\infty\) as \(x\) tends to \(+\infty\). When \(x=0, a^{x}=1\); and when \(x\) tends to \(-\infty\), the function tends to \(0\) . The general graph of the function looks like as shown here, where \(a=2\).
	Case 2: \(0<a<1\) The function decreases rapidly with increasing \(x\). The function tends to \(0\) as \(x\) tends to \(+\infty\). When \(x=0, a^{x}=1\) as usual; and when \(x\) tends to \(-\infty\), the function tends to \(+\infty\). The general graph of such a function looks as shown here, where \(a=2\).

Exponential Functions vs Logarithmic Functions

The logarithmic and the exponential functions are inverses of each other. The logarithmic function does the opposite of taking the power of a number. Let’s look at it mathematically.

Exponential Form: \(a^{y}=x\)

Logarithmic form: \(y=\log _{a} x\)

With these two forms, you can easily see that the value of the function \(f(x)=\log _{a} x\) is the power to which \(a\) must be raised to get \(x\). Therefore, \(x\) cannot be negative since that would require \(a\) to be imaginary. The conditions on the base \(a\) are:

• \(a>0\) : It follows directly from the exponential representation of the logarithmic function.
• \(\quad \boldsymbol{a} \neq 1:\) Since \(1\) raised to any power would only give \(1\) .

Cases of Logarithmic Functions

Based on the value of \(a\), we have two possible cases for logarithmic functions:

Case 1: \(a>1\) The logarithmic function increases slowly with increasing \(x\). The function tends to \(-\infty\) as \(x\) tends to \(0\). When \(x\) tends to \(+\infty\), the function also tends to \(+\infty\) The general graph of the function looks as shown here, where \(a=2\).
	Case 2: \(0<a<1\) The function increases to \(+\infty\) as \(x\) tends to \(0\). The function falls at a decreasing rate to \(-\infty\) as \(x\) tends to \(+\infty\). The general graph is as shown here, where \(a=0.5\).

What are the Limits of a Function?

A limit of a function is usually represented as

\(\mathop {\lim }\limits_{x \to c} f(x) = L\)

This is read as, “The limit of \(f(x)\) as \(x\) approaches \(c\) equals \(L.”\)

Like any other function, exponential and logarithmic functions have limits. Then tend to obey specific end behaviours when the value of \(x\) tends to \(\pm \infty\).

The limit of \(f\) at \(x=a\), is the value \(f\) approaches as we get closer to \(x=a\). Graphically, this is the \(y\)-value we approach when we look at the graph of \(f\) and get closer to the point \(x= a\) on the graph.

Limits of Exponential Functions

The limit of the exponential function can be easily determined from their graphs.

From the graph of the exponential function, \(a^{x}\), where \(a>1\), we can see that the graph is increasing. It is an increasing function.
When \(x \rightarrow-\infty\), the graph of \(f(x)=a^{x}\) tends to \(0\) .
\(y=0\) is said to be the horizontal asymptote.
When \(x \rightarrow+\infty\), the graph of \(f(x)=a^{x}\) tends to \(+\infty\).

From the graph of the exponential function \(a^{x}\), when \(0<a<1\), we can see that the graph is decreasing. It is a decreasing function.

When \(x \rightarrow-\infty\), the graph of \(f(x)=a^{x}\) tends to \(+\infty\).

When \(x \rightarrow+\infty\), the graph of \(f(x)=a^{x}\) tends to \(0\).

\(y=0\) is said to be the horizontal asymptote.

When considering the natural exponential function, then as \(e=2.71828\), the condition that \(a>1\) only persists.

Hence, we can define the limit of \(e^{x}\) for various conditions as below:

• \(\mathop {\lim }\limits_{x \to \infty } {e^x} = \infty \)
• \(\mathop {\lim }\limits_{x \to \infty } {e^x} = 0\)
• \(\mathop {\lim }\limits_{x \to \infty } {e^{ – x}} = 0\)
• \(\mathop {\lim }\limits_{x \to \infty } {e^{ – x}} = \infty \)

Limits of Logarithmic Functions

The logarithmic limits can be easily understood from their graphs.

From the graph of the logarithmic function \(\log x\), where \(a>1\), we can see that the graph is increasing. It is an increasing function.

When \(x \rightarrow-\infty\), the graph of \(f(x)=a^{x}\) tends to \(0\) .

When \(x \rightarrow+\infty\), the graph of \(f(x)=a^{x}\) tends to \(+\infty\).

\(x=0\) is said to be the horizontal asymptote.

From the graph of the logarithmic function, \(\log x\), when \(0<a<1\), we can see that the graph increases. It is an increasing function.

When \(x \rightarrow-\infty\), the graph of \(f(x)=a^{x}\) tends to \(0\) .

When \(x \rightarrow+\infty\), the graph of \(f(x)=a^{x}\) tends to \(+\infty\).

\(x=0\) is said to be the horizontal asymptote.

The natural logarithm is the \(\log\) to the base \(e\) is represented as \(\ln\).

The limits of the natural logarithm functions can be defined as below:

• \(\mathop {\lim }\limits_{x \to {0^ + }} \ln x = – \infty \)
• \(\mathop {\lim }\limits_{x \to \infty } \ln x = \infty \)

Formulas of Exponential and Logarithmic Limits

Listed below are the limits involved with exponential and logarithmic functions:

1. \(\mathop {\lim }\limits_{x \to \infty } {b^x} = \infty \) if \(b>1\)

2. \(\mathop {\lim }\limits_{x \to – \infty } {b^x} = 0\) if \(b>1\)

3. \(\mathop {\lim }\limits_{x \to \infty } {b^x} = 0\) if \(0<b<1\)

4. \(\mathop {\lim }\limits_{x \to – \infty } {b^x} = \infty \), if \(0<b<1\)

5. \(\mathop {\lim }\limits_{x \to 0} {(1 + x)^{\frac{1}{x}}} = e\)

6. \(\mathop {\lim }\limits_{x \to 0} {\left( {1 + \frac{1}{x}} \right)^x} = e\)

7. \(\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} – 1}}{x} = 1\)

8. \(\mathop {\lim }\limits_{x \to 0} \frac{{\log (1 + x)}}{x} = 1\)

Solved Examples

Below are a few solved examples that can help in getting a better idea:

Q.1. Find the limit of the function \(f(x)=3^{x}\) when \(x\) tends to \(\infty\).
Sol: From the graph of the exponential function \(a^{x}\), when \(a>1\), we can see that the graph is increasing.
When \(x \rightarrow+\infty\), the graph of \(f(x)=a^{x}\) tends to \(+\infty\).
Hence, \(\mathop {\lim }\limits_{x \to \infty } {3^x} = \infty \)

Q.2. Find the limit of the exponential function \(\frac{e^{3 x}-1}{3 x}\).
Sol: We know that \(\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} – 1}}{x} = 1\)
Hence, \(\mathop {\lim }\limits_{x \to 0} \frac{{{e^{3x}} – 1}}{{3x}} = 3\)

Q.3. Find the limit of \(\mathop {\lim }\limits_{x \to \infty } {e^{2 – 4x – 8{x^2}}}\)
Sol: Let us first find the limit of the power of the exponential function.
\(\mathop {\lim }\limits_{x \to \infty } 2 – 4x – 8{x^2} = – \infty \)
Hence, the limit will become,
\(\mathop {\lim }\limits_{x \to \infty } {e^{2 – 4x – 8{x^2}}} = \mathop {\lim }\limits_{x \to \infty } {e^{ – \infty }}\)
\(=0\)

Q.4. Find the value of \(\mathop {\lim }\limits_{x \to \infty } \frac{{6{e^{4x}} – {e^{ – 2x}}}}{{8{e^{4x}} – {e^{2x}} + 3{e^{ – x}}}}\)
Sol: Given \(\mathop {\lim }\limits_{x \to \infty } \frac{{6{e^{4x}} – {e^{ – 2x}}}}{{8{e^{4x}} – {e^{2x}} + 3{e^{ – x}}}} = \mathop {\lim }\limits_{x \to \infty } \frac{{{e^{4x}}\left( {6 – {e^{ – 6x}}} \right)}}{{{e^{4x}}\left( {8 – {e^{ – 2x}} + 3{e^{ – 5x}}} \right)}}\)
\(\mathop {\lim }\limits_{x \to \infty } \frac{{{e^{4x}}\left( {6 – {e^{ – 6x}}} \right)}}{{{e^{4x}}\left( {8 – {e^{ – 2x}} + 3{e^{ – 5x}}} \right)}}\)
\(=\frac{6-0}{8-0+0}\)
\(=\frac{3}{4}\)
Hence, the value of \(\mathop {\lim }\limits_{x \to \infty } \frac{{6{e^{4x}} – {e^{ – 2x}}}}{{8{e^{4x}} – {e^{2x}} + 3{e^{ – x}}}}\) is \(\frac{3}{4}\)

Q.5. Find the value of \(\mathop {\lim }\limits_{x \to \infty } \ln \left( {7{x^3} – {x^2} + 1} \right)\)
Sol: Let us first find the limit of the argument of the logarithmic function.
\(\mathop {\lim }\limits_{x \to \infty } 7{x^3} – {x^2} + 1 = \infty \)
So, now the given limit change as
\(\mathop {\lim }\limits_{x \to \infty } \ln \left( {7{x^3} – {x^2} + 1} \right) = \mathop {\lim }\limits_{x \to \infty } \ln \left( \infty \right)\)
\(=\infty\)

Summary

An exponential function is \(f(x)=b^{y}\). The logarithmic function is given by \(f(x)=\log _{b} x=\)\(y\). When \(a>1, f(x)\) increases with increasing \(x\) and tends to \(+\infty\) as \(x\) tends to \(+\infty\), and when \(x\) tends to \(-\infty\), the function tends to \(0\) . When \(a<1\), the function decreases with increasing \(x\) and tends to \(0\) as \(x\) tends to \(+\infty\), and when \(x\) tends to \(-\infty\), the function tends to \(+\infty\).

When \(a>1\), the logarithmic function increases with increasing \(x\) and tends to \(-\infty\) as \(x\) tends to \(0\) . When \(x\) tends to \(+\infty\), the function also tends to \(+\infty\). When \(0<a<1\), the function increases to \(+\infty\) as \(x\) tends to \(0\) and falls to \(-\infty\) as \(x\) tends to \(+\infty\).

FAQs of Exponential and Logarithmic Limits

Students might be having many questions regarding the Exponential and Logarithmic Limits. Here are a few commonly asked questions and answers:

Q.3. What is natural logarithm?
Ans: The natural logarithm is the log to the base \(e\) is represented as \(\ln\).

Q.4. What is the difference between the exponential and logarithmic functions?
Ans: If the function is of the form \(f(x)=b^{y}\), where \(b>0\) and \(b \neq 1\), then it is an exponential function. The quantity \(x\) is the number, \(b\) is the base, and \(y\) is the exponent or power. If the function is of the form \(f(x)=\log _{b} x=y\), it is a logarithmic function. Here \(b\) is the base, \(y\) is the exponent, and \(x\) is the argument.

Q.5. What is the limit of \(\ln x\) as \(x\) approaches \(0\) ?
Ans: The limit of \(\ln x\) as \(x\) approaches zero does not exist. For a limit to exist, both the left-hand and right-hand limits must exist, and be of the same value. But, here \(\mathop {\lim }\limits_{x \to {0^ – }} \ln x\) does not exist, and hence the limit does not exist.

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