Understanding the Limit of a Function as x Approaches Infinity | Explained with an Example

lim x->∞ pow top > pow bottom

The notation “lim x->∞ pow top > pow bottom” refers to the limit of a function as x approaches infinity, where the function involves a power (exponent) of the top and bottom

The notation “lim x->∞ pow top > pow bottom” refers to the limit of a function as x approaches infinity, where the function involves a power (exponent) of the top and bottom.

To explain this in more detail, let’s consider an example. Suppose we have a function f(x) = (x^2 + 3x)/(2x – 5). Here, the “top” is represented by x^2 + 3x, and the “bottom” is represented by 2x – 5. In this case, we are interested in finding the limit of f(x) as x approaches infinity.

To evaluate this limit, we can simplify the function by dividing both the top and bottom by the highest power of x. In this case, x^2 has the highest power of x. So dividing both the top and bottom by x^2, we get:

f(x) = (x^2 + 3x)/(2x – 5) = (1 + 3/x)/(2/x – 5/x^2)

Now, as x approaches infinity, it means that the value of x is becoming larger and larger. As a result, the terms involving 3/x and 5/x^2 become negligible compared to the term 1/x^2.

So, taking the limit as x approaches infinity:

lim x->∞ (1 + 3/x)/(2/x – 5/x^2)

The term 3/x approaches 0, as x becomes larger and larger. Similarly, the term 5/x^2 also approaches 0. Thus, we are left with:

lim x->∞ (1 + 0)/(0 – 0)

Note that we obtained this result by considering the highest power terms and their behavior as x approaches infinity. In this case, both the top and bottom became 0, but the expression remains indeterminate.

To evaluate the limit further, we can use L’Hôpital’s rule or other techniques depending on the specific function. L’Hôpital’s rule states that if we have an indeterminate form of 0/0 or ∞/∞, we can differentiate the top and bottom functions with respect to x and then take the limit again.

In this example, applying L’Hôpital’s rule:

lim x->∞ (1 + 3/x)/(2/x – 5/x^2) = lim x->∞ (0)/(0) = Indeterminate form

If the function allows further simplification or a different approach, it may be possible to find a finite limit. However, in this case, the limit remains indeterminate, meaning we cannot determine a specific value for it solely based on the given expression. Further analysis or different techniques may be required to obtain a definite answer.

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