Limit of a 0 x infinity Indeterminate Form | Applying Mathematical Techniques to Find the Limit

To find the limit of a 0 x infinity indeterminate form…

To find the limit of a 0 x infinity indeterminate form, you can apply various mathematical techniques

To find the limit of a 0 x infinity indeterminate form, you can apply various mathematical techniques. One common approach is to rewrite the expression in a different form that allows you to evaluate the limit more easily.

One method is to convert the problem into a limit that involves the indeterminate form 0/0. To achieve this, you can manipulate the expression and apply algebraic techniques. Let’s consider an example:

Suppose we want to find the limit of f(x) as x approaches infinity, where f(x) = x * e^(-x).

Step 1:
Start by taking the logarithm of both sides of the function to obtain ln(f(x)) = ln(x) + ln(e^(-x)), which simplifies to ln(f(x)) = ln(x) – x.

Step 2:
Now rewrite the function in a different form by using the exponential function. Taking the exponential of both sides gives e^(ln(f(x))) = e^(ln(x) – x). This further simplifies to f(x) = e^(ln(x)) * e^(-x) = x * e^(-x).

Step 3:
At this point, you have converted the original expression into the 0/0 indeterminate form. Now you can differentiate both sides with respect to x. Differentiating f(x) gives f'(x) = 1 * e^(-x) – x * e^(-x), which becomes f'(x) = e^(-x) – x * e^(-x).

Step 4:
To evaluate the limit, set the derivative f'(x) equal to zero and solve for x to find the critical point. In this case, the critical point occurs when e^(-x) – x * e^(-x) = 0, which simplifies to e^(-x)(1 – x) = 0. As e^(-x) is never zero, the critical point is x = 1.

Step 5:
Now that you have the critical point, you can compare the behavior of the function on both sides of x = 1. For x < 1, notice that f'(x) < 0, indicating the function is decreasing. For x > 1, f'(x) > 0, indicating the function is increasing. Thus, you can conclude that f(x) has a maximum value at x = 1 and that the limit of f(x) as x approaches infinity is 0.

Therefore, using the technique outlined above, the limit of f(x) as x approaches infinity, where f(x) = x * e^(-x), is 0.

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