Approaching from the right

Definition

Approaching from the right refers to evaluating the limit of a function as the input values approach a particular point from values greater than that point. This method is crucial for understanding one-sided limits and can reveal different behaviors of functions at points where they may not be defined or behave differently compared to when approaching from the left.

5 Must Know Facts For Your Next Test

1. When approaching from the right, you look at values just greater than the target point to determine the limit.
2. If a function has a different limit when approached from the right versus from the left, it is considered to have no overall limit at that point.
3. This concept is essential for functions with discontinuities or piecewise definitions, where behavior can change significantly at specific points.
4. Graphically, approaching from the right means tracing along the curve of the function starting just to the right of the point in question.
5. In many cases, evaluating limits by approaching from the right can help determine vertical asymptotes and other critical points in functions.

Review Questions

• How does approaching from the right differ from approaching from the left in evaluating limits?
  • Approaching from the right means considering values that are greater than a certain point while evaluating limits, while approaching from the left involves values that are less than that point. This difference is important because a function may behave differently depending on which side you approach it from, leading to distinct one-sided limits. If these limits are not equal, it indicates that there is no overall limit at that point.
• Explain why approaching from the right is significant for identifying discontinuities in a function.
  • Approaching from the right is significant for identifying discontinuities because it allows us to analyze how a function behaves specifically at points where it may not be defined or has sudden changes. For example, if we find that the limit approaching from the right does not match the limit from the left, it suggests a jump discontinuity or an infinite discontinuity. Understanding these behaviors helps us better grasp how functions operate and where they may fail to be continuous.
• Evaluate and compare the limits of a piecewise function as $x$ approaches 2 from both sides and discuss what this reveals about its continuity.
  • To evaluate a piecewise function at $x = 2$, we need to find both $ ext{lim}_{x o 2^-} f(x)$ and $ ext{lim}_{x o 2^+} f(x)$. If, for example, $ ext{lim}_{x o 2^-} f(x) = 3$ and $ ext{lim}_{x o 2^+} f(x) = 5$, we notice that these two limits do not match. This discrepancy indicates that there is a jump discontinuity at $x = 2$, revealing that while the function may be defined at this point, it does not meet the criteria for continuity because both one-sided limits must equal each other and also equal $f(2)$ for continuity to hold.

Related terms

One-sided limit: A limit that considers only values approaching a certain point from one side, either from the left or the right.

Limit:

The value that a function approaches as the input approaches a specified value.

Continuity:

A property of a function where it is uninterrupted at a point, meaning the limit from both sides equals the function's value at that point.