Calculating Limit as x Approaches Infinity

In summary, the concept of calculating limit as x approaches infinity is an important concept in calculus that involves determining the behavior of a function as the input approaches infinity. To calculate this limit, one must evaluate the function at increasingly large values of x. There is a difference between one-sided and two-sided limits, with the latter considering the behavior on both sides of the input value. It is possible for a function to have a limit as x approaches infinity but not at a specific value, which is often seen with functions that have vertical asymptotes. Real-world applications of this concept include finding maximum or minimum values, predicting behavior of systems, and analyzing phenomena in various fields.
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I'm trying to find $\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x}$

$\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x} =\lim_{x \to 20^{+}}\frac{5+1/x^3}{20-8000/x^2} = \frac{5+\lim_{x \to 20^{+}}1/x^3}{20-\lim_{x \to 20^{+}}8000/x^2} = \frac{5+\frac{1}{8000}}{20-\frac{8000}{400}} = \infty. $

I'm not sure because it seems I have a zero dominator throughout.
 
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$\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x}$

Since $5\cdot 20^3+1>0, 20 \cdot 20^3-8000 \cdot 20=0$ and we approach $20$ from the right side, we can immediately say that $\displaystyle \lim_{x \to 20^{+}}\frac{5x^3+1}{20x^3-8000x}=+\infty$.
 

FAQ: Calculating Limit as x Approaches Infinity

What is the concept of calculating limit as x approaches infinity?

The concept of calculating limit as x approaches infinity is a fundamental concept in calculus. It involves determining the behavior of a function as the input (x) approaches infinity, or the value of the function when the input is infinitely large.

How do you calculate the limit as x approaches infinity?

To calculate the limit as x approaches infinity, you need to evaluate the function at increasingly large values of x. If the function approaches a finite number as x gets larger and larger, then that number is the limit. If the function increases or decreases without bound, then the limit does not exist.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit is a limit where x approaches a specific value from only one direction (either positive or negative). A two-sided limit is a limit where x approaches a specific value from both positive and negative directions. In other words, a two-sided limit considers the behavior of the function on both sides of the input value, while a one-sided limit only considers the behavior on one side.

Can a function have a limit as x approaches infinity but not at a specific value?

Yes, it is possible for a function to have a limit as x approaches infinity but not at a specific value. This means that the function is approaching a finite number as x gets larger, but it may not have a defined value at a specific point. This is often the case for functions with vertical asymptotes, where the function approaches infinity at a certain point but does not have a defined value at that point.

What are some real-world applications of calculating limit as x approaches infinity?

Calculating limit as x approaches infinity has many real-world applications, such as determining the maximum or minimum values of a function, finding the rate of change of a function, and predicting the behavior of systems over time. It is also used in fields such as physics, engineering, and economics to model and analyze real-world phenomena.

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