15.9. Limit of Exponential and Logarithmic Functions
Exponential Functions
An exponential function is defined as:
Limit of an Exponential Function
Exponential functions are continuous over the set of real numbers with no jump or hole discontinuities. To evaluate the limit of an exponential function, plug in the value of c.
Illustrative Example
Find the limit of the exponential function below.
Illustrative Example
Find the limit of the exponential function below.
To find the limit, simplify the expression by plugging in 1: 3^{ 2 ( 1 ) - 1 } = 3. The limit is 3.
Calculator solution
Type in: lim [ x = 1 ] ( 3 ^ ( 2x - 1 ) )
Type in: lim [ x = 1 ] ( 3 ^ ( 2x - 1 ) )
More Examples
Calculate each limit if it exists.
Calculate each limit if it exists.
Calculator solution
Type in: lim [ x = -2 ] ( 3^( 2x - 1 ) )
Type in: lim [ x = -2 ] ( 3^( 2x - 1 ) )
Calculator solution
Type in: lim [ x = 4 + ] ( 3^( 2x - 1 ) )
Type in: lim [ x = 4 + ] ( 3^( 2x - 1 ) )
Calculator solution
Type in: lim [ x = 4+ ] ( 1 / 2^( 2x - 1 ) + 5 ) )
Type in: lim [ x = 4+ ] ( 1 / 2^( 2x - 1 ) + 5 ) )
Calculator solution
Type in: lim [ x = 3 + ] ( 1 / 2^( 2x - 1 ) - 5 ) )
Type in: lim [ x = 3 + ] ( 1 / 2^( 2x - 1 ) - 5 ) )
Calculator solution
Type in: lim [ x = π/2 ] ( 2^sin x )
Type in: lim [ x = π/2 ] ( 2^sin x )
Calculator solution
Type in: lim [ x = 3 ] ( e^(x - 1) ) + 3
Type in: lim [ x = 3 ] ( e^(x - 1) ) + 3
Calculator solution
Type in: lim [ x = -2 ] ( 2e^( x + 2 ) - 3 )
Type in: lim [ x = -2 ] ( 2e^( x + 2 ) - 3 )
Logarithmic Functions
A logarithmic function can be written in any of the following forms:
The first form has base b such that b > 0, b ≠ 1. The base of the second equation is understood to be 10. The third equation has base e. In all three forms, x > 0.
Since a logarithmic function is the inverse of an exponential function, it is also continuous. Therefore the limit as x approaches c can be similarly found by plugging c into the function.
Illustrative Example
Find the limit of the logarithmic function below.
Since a logarithmic function is the inverse of an exponential function, it is also continuous. Therefore the limit as x approaches c can be similarly found by plugging c into the function.
Illustrative Example
Find the limit of the logarithmic function below.
Solution
1) Plug x = 3 into the expression ( 3x - 5 )
3(3) - 5 = 4
2) Evaluate the logarithm with base 4.
Since 4^1 = 4, the value of the logarithm is 1.
3) The limit as x approaches 3 is 1.
Calculator solution
Type in: lim [ x = 3 ] log[4]( 3x - 5 )
1) Plug x = 3 into the expression ( 3x - 5 )
3(3) - 5 = 4
2) Evaluate the logarithm with base 4.
Since 4^1 = 4, the value of the logarithm is 1.
3) The limit as x approaches 3 is 1.
Calculator solution
Type in: lim [ x = 3 ] log[4]( 3x - 5 )
More Examples
Calculate the limit of each logarithm if it exists.
Calculator solution
Type in: lim [ x = 2 ] ( log [2] ( 3x - 5 ) + 2 ) )
Type in: lim [ x = 2 ] ( log [2] ( 3x - 5 ) + 2 ) )
Calculator solution
Type in: lim [ x = 3 ] ( -root[5] x + e^x / ( 1 + lnx ) )
Type in: lim [ x = 3 ] ( -root[5] x + e^x / ( 1 + lnx ) )
Calculator solution
Type in: lim [ x = 3 ] log ( 3x - 5 )
Type in: lim [ x = 3 ] log ( 3x - 5 )
Calculator solution
Type in: lim [ x = 1 ] log [5] ( x + 6 )^2
Type in: lim [ x = 1 ] log [5] ( x + 6 )^2