Use a graphing utility to complete the table and estimate the limit as [tex]$x$[/tex] approaches infinity. (Round your answers to five decimal places.)
[tex]$f(x)=x+\left(\frac{1}{6 x}\right)$[/tex]
| x | [tex]$10^0$[/tex] | [tex]$10^1$[/tex] | [tex]$10^2$[/tex] | [tex]$10^3$[/tex] | [tex]$10^4$[/tex] | [tex]$10^5$[/tex] | [tex]$10^6$[/tex] |
|---|---|---|---|---|---|---|---|
| f(x) | | | | | | |
Use a graphing utility to graph the function and estimate the limit. Find the limit analytically and compare your results with the estimates. What is the exact limit? (If an answer does not exist, enter DNE.)
[tex]$\lim _{x \rightarrow \infty} f(x)=[/tex]
Answer (1)
Calculate f ( x ) for x = 1 , 10 , 100 , 1000 , 10000 , 100000 , 1000000 , observing that f ( x ) gets closer to x as x increases.
Estimate lim x → ∞ f ( x ) as ∞ based on the table.
Analytically determine lim x → − ∞ f ( x ) = lim x → − ∞ ( x + 6 x 1 ) .
Conclude that lim x → − ∞ f ( x ) = − ∞ , since lim x → − ∞ x = − ∞ and lim x → − ∞ 6 x 1 = 0 , so the final answer is − ∞ .
Explanation
Problem Analysis We are given the function f ( x ) = x + 6 x 1 and asked to complete a table of values, estimate the limit as x approaches infinity using the table and a graphing utility, and find the limit as x approaches negative infinity analytically.
Completing the Table First, let's complete the table using the given x values: 1 0 0 , 1 0 1 , 1 0 2 , 1 0 3 , 1 0 4 , 1 0 5 , 1 0 6 . We will calculate f ( x ) for each of these values.
Calculating f(1) For x = 1 0 0 = 1 , f ( 1 ) = 1 + 6 ( 1 ) 1 = 1 + 6 1 = 1 + 0.16667 = 1.16667
Calculating f(10) For x = 1 0 1 = 10 , f ( 10 ) = 10 + 6 ( 10 ) 1 = 10 + 60 1 = 10 + 0.01667 = 10.01667
Calculating f(100) For x = 1 0 2 = 100 , f ( 100 ) = 100 + 6 ( 100 ) 1 = 100 + 600 1 = 100 + 0.00167 = 100.00167
Calculating f(1000) For x = 1 0 3 = 1000 , f ( 1000 ) = 1000 + 6 ( 1000 ) 1 = 1000 + 6000 1 = 1000 + 0.00017 = 1000.00017
Calculating f(10000) For x = 1 0 4 = 10000 , f ( 10000 ) = 10000 + 6 ( 10000 ) 1 = 10000 + 60000 1 = 10000 + 0.00002 = 10000.00002
Calculating f(100000) For x = 1 0 5 = 100000 , f ( 100000 ) = 100000 + 6 ( 100000 ) 1 = 100000 + 600000 1 = 100000 + 0.00000 = 100000.00000
Calculating f(1000000) For x = 1 0 6 = 1000000 , f ( 1000000 ) = 1000000 + 6 ( 1000000 ) 1 = 1000000 + 6000000 1 = 1000000 + 0.00000 = 1000000.00000
Estimating the Limit as x Approaches Infinity As x approaches infinity, the term 6 x 1 approaches 0, so f ( x ) approaches x . Therefore, lim x → ∞ f ( x ) = ∞ .
Finding the Limit as x Approaches Negative Infinity Analytically Now, let's find the limit as x approaches negative infinity analytically: lim x → − ∞ f ( x ) = lim x → − ∞ ( x + 6 x 1 ) = lim x → − ∞ x + lim x → − ∞ 6 x 1 . Since lim x → − ∞ x = − ∞ and lim x → − ∞ 6 x 1 = 0 , then lim x → − ∞ ( x + 6 x 1 ) = − ∞ + 0 = − ∞ .
Final Answer The exact limit as x approaches negative infinity is − ∞ .
Examples
In physics, this function could represent the position of an object where 'x' is time and 1/(6x) is a damping factor that diminishes over time. Understanding the limit as x approaches infinity helps predict the object's long-term position. Similarly, in engineering, it could model a system's output where 'x' is the input signal strength, and the fractional term represents a diminishing error. Analyzing such limits helps engineers design systems that stabilize over time, ensuring reliable performance even with varying inputs.
