91Ó°ÊÓ

When studying piecewise functions like \( f(x) \), the concept of limits helps us understand the behavior of the function as \( x \) approaches certain values or boundary points. In our function \( f(x) \), as \( x \) approaches 1 from the left (or from values less than 1), we look at the expression \( \frac{1}{1-x} \). This expression tends to infinity because the denominator \( 1-x \) becomes very small when \( x \) is near 1. Thus, the limit of \( f(x) \) as \( x \) approaches 1 from the left is \( \infty \).
This behavior shows us an essential aspect of piecewise functions: they can behave differently on different intervals.

• The limit tests how functions approach a value but does not require them to ever actually reach it.
• Understanding the limit as \( x \) nears 1 tells us that \( f(x) \) increases without bound and therefore doesn’t have a maximum within \([0,1)\).

Recognizing this kind of unbounded behavior is vital for understanding why certain functions can lack a maximum value.

Continuity in functions refers to the characteristic of a function to be unbroken or seamless within a given interval. A function is continuous if, as you hone in on any point in its domain, the approaches from both directions align at the same function value. In the context of our piecewise function, the continuity breaks at \( x=1 \).
On the interval \( 0 \leq x < 1 \), the function \( \frac{1}{1-x} \) is continuous itself. However, at \( x=1 \), the function jumps to a value of 0, creating a discontinuity.

• This discontinuity is because the left-hand limit as \( x \) approaches 1 tends towards infinity, while the actual function value at \( x = 1 \) is specified as 0.
• The presence of different functional rules within different parts of the domain leads to discontinuity at the boundary point.

Evaluating the continuity is crucial since it impacts the possible range of a function on a closed interval.

Evaluating piecewise functions at their boundaries can be quite insightful for determining characteristics like minimums and maximums. In the exercise, we examined the boundary at \( x=0 \) and \( x=1 \) for the function \( f(x) \). At \( x=0 \), we calculate \( f(0) = \frac{1}{1-0} = 1 \). This gives us a point at which the function is well-defined and finite.
At \( x=1 \), \( f(1) = 0 \) serves as the pinpointing of the function on the closed boundary.

• Since the value at the boundary \( x=1 \) is the lowest value within the interval, it marks the minimum.
• No highest defined boundary or point within the interval keeps the function from having a maximum, as it approaches infinity instead.

Assessing boundaries helps us understand fully how the piecewise function behaves from start to finish of the given interval.