91Ó°ÊÓ

Limit evaluation is a technique used in calculus to determine the behavior of a function as the input approaches a specified value. In our exercise, we are evaluating the limit of the function \( f(x) = \frac{2}{1-x} \) as \( x \rightarrow 1^- \). This means we need to explore what happens to the function when \( x \) gets very close to 1, but is slightly less than 1. To evaluate the limit, you can use different methods, like:

• Graphically: By observing the graph of the function to see the value it approaches as \( x \) approaches the desired point.
• Analytically: By manipulating the function algebraically to find the behavior as \( x \) nears the limit point.
• Numerically: By computing the function at values approaching the limit point and observing the trend.

In our problem, we use a numerical approach, substituting values of \( x \) that are close to 1 from the left side (like 0.9, 0.99, 0.999) into \( f(x) \) and examining the results. This helps us determine how \( f(x) \) behaves as the input approaches 1.

When discussing limits, "approaching from the left" refers to observing what happens to a function as the input approaches a particular value from values less than that number. For our function \( f(x) = \frac{2}{1-x} \), we consider \( x \rightarrow 1^- \). This implies using values of \( x \) slightly less than 1 to probe the behavior.Why do we use a left-hand approach? Sometimes a function behaves differently when approaching a value from the left versus approaching from the right. The notation \( 1^- \) specifically directs us to use values:
- For instance, \( x = 0.9, 0.99, 0.999 \).By choosing these values and calculating \( f(x) \), we can see how the function behaves as \( x \) gets very close to 1 from the left. This process allows us to understand the function's behavior enough to determine the left-sided limit, which can be crucial if the function has a discontinuity at the point of interest.

Infinity in limits is a concept that arises when the output of a function increases or decreases without bound as the input approaches a certain point. In our example, the function \( f(x) = \frac{2}{1-x} \) increases without bound as \( x \) approaches 1 from the left.When we calculated \( f(x) \) for values like:

• \( f(0.9) = 20 \)
• \( f(0.99) = 200 \)
• \( f(0.999) = 2000 \)
• \( f(0.9999) = 20000 \)

We noticed that the function values grow larger and larger. This suggests that as \( x \) approaches 1 from the left, the function tends towards positive infinity. Infinity in limits signifies that there's no finite bound for the values the function can take as \( x \) approaches the particular point. Instead of approaching a specific number, the trend points towards a boundless increase or decrease. This concept is crucial for understanding behaviors near vertical asymptotes or points of discontinuity in functions.