Investigate the limit numerically and graphically.
lim𝑥→±∞16𝑥+14𝑥2+9⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√
limx→±∞⁡16⁢x+14⁢x2+9

Calculate the values of 𝑓(𝑥)=16𝑥+14𝑥2+9⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√f⁡(x)=16⁢x+14⁢x2+9 for 𝑥=±100,x=±100, ±500,±500, ±1000,±1000, and ±10000.and ±10000.

(Use decimal notation. Give your answers to six decimal places.)
𝑓(−100)=f⁡(−100)=
𝑓(−500)=f⁡(−500)=
𝑓(−1000)=f⁡(−1000)=
𝑓(−10000)=f⁡(−10000)=
𝑓(100)=f⁡(100)=
𝑓(500)=f⁡(500)=
𝑓(1000)=f⁡(1000)=
𝑓(10000)=f⁡(10000)=
Graph 𝑓(𝑥)f⁡(x) using the graphing utility.
𝑓(𝑥)=f⁡(x)=


The graph of \( f(x) = \sqrt{16x + 14x^2 + 9} \) shows how the function behaves as \( x \) approaches both positive and negative infinity. As \( x \) grows larger, \( f(x) \) increases steadily, and the function appears to become more linear at large values of \( x \).

Would you like more details on this, or do you have any questions?

Here are 5 related questions:
1. How do the asymptotic properties of the function change as \( x \to \infty \) and \( x \to -\infty \)?
2. What is the behavior of \( f(x) \) near \( x = 0 \)?
3. How does this function compare to simpler functions like \( \sqrt{14x^2} \)?
4. What would happen if the coefficients of \( x^2 \) or \( x \) were changed?
5. How can the graph of \( f(x) \) help predict limits analytically?

**Tip:** To find limits as \( x \to \infty \) or \( x \to -\infty \), analyzing the dominant terms in the expression helps greatly.

Investigate the limit numerically and graphically. lim(x→±∞) sqrt(16x + 14x^2 + 9). Calculate the values of f(x) = sqrt(16x + 14x^2 + 9) for x = ±100, ±500, ±1000, and ±10000. (Use decimal notation. Give your answers to six decimal places.) Graph f(x) using the graphing utility.

Learn how to investigate the limit of the function f(x) = sqrt(16x + 14x^2 + 9) numerically and graphically as x approaches positive and negative infinity. This problem covers concepts of limits, functions, square roots, and asymptotic behavior, suitable for students in Grades 10-12.

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