91Ó°ÊÓ

When we talk about one-sided limits, we're focusing on understanding how a function behaves as it approaches a particular point from one side only. Essentially, we look at two kinds of one-sided limits: the right-hand limit and the left-hand limit.

• The right-hand limit (\( ext{lim}_{x o a^+} f(x) \)) examines the function values as they get closer to a specific point of interest, \( a \), from values greater than \( a \).
• The left-hand limit (\( ext{lim}_{x o a^-} f(x) \)) looks at the function values approaching \( a \) from values less than \( a \).

In our exercise, it was crucial to evaluate both these limits as \( x \) approaches 1 to understand the behavior of the given expression. For example, as \( x \) approaches 1 from the right, we handled the function differently than when \( x \) approached 1 from the left. This difference is key to determining the overall behavior and nature of the limit at a point.

The absolute value function is a fundamental concept that measures the distance of a number from zero on the number line. It ensures non-negative results regardless of whether the input is positive or negative.
In mathematical terms, the absolute value of a number \( x \) is denoted as \( |x| \). It is defined as follows:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

In our exercise, we saw how the expression inside the absolute value, \( |x-1| \), behaved differently depending on whether \( x \) was greater or less than 1. For \( x \) approaching 1 from the right (\( x > 1 \)), \( |x-1| \) was simply \( x-1 \). In contrast, for \( x \) approaching 1 from the left (\( x < 1 \)), we had \( |x-1| = -(x-1) \). This intricacy played a crucial role in how the expression was simplified for each one-sided limit.

The square root function is a mathematical operation that returns the number which, when multiplied by itself, results in the given number. Symbolized as \( \sqrt{x} \), it always returns the non-negative root of a real number, with a few specific properties:

• The function is only defined for non-negative inputs (\( x \geq 0 \)).
• For each non-negative number, there is exactly one square root.

In the context of our exercise, the function \( \sqrt{2x} \) was pivotal. As \( x \) approached 1, \( \sqrt{2x} \) approached \( \sqrt{2} \). The function's smooth and continuous nature provided straightforward calculations for the limit problems. It demonstrated how square roots can simplify mathematical expressions when evaluated within their domain.

A two-sided limit explores the behavior of a function as the input approaches a particular point from both directions. It's about how the function's "output" behaves as \( x \) approaches \( a \) from both the left (\( x o a^- \)) and the right (\( x o a^+ \)). For a two-sided limit to exist at a point \( a \), the left-hand limit and the right-hand limit must both exist and must be equal:

• If \( ext{lim}_{x o a^-} f(x) = ext{lim}_{x o a^+} f(x) = L \), then \( ext{lim}_{x o a} f(x) = L \).

In our example, the limits from both sides, approaching 1, were not equal. On one side, we found \( ext{lim}_{x o 1^+} f(x) = \sqrt{2} \), while on the other side, \( ext{lim}_{x o 1^-} f(x) = -\sqrt{2} \). This disparity means that the two-sided limit does not exist at \( x = 1 \). Hence, understanding both one-sided and two-sided limits gives a complete picture of how the function behaves around the point of interest.