How to Spot One-Sided Limits from a Graph

Understanding one-sided limits is important in calculus and helps us study how functions work. There are two main types of one-sided limits to look at when we want to see how a function behaves as it gets close to a certain point: left-hand limits and right-hand limits.

1. What Are One-Sided Limits?

• Left-hand Limit: This tells us what happens to a function ( f(x) ) as ( x ) gets close to a number ( a ) from the left side (the smaller values). We write it as $\lim_{x \to a^-} f(x)$.

• Right-hand Limit: This shows us what happens to ( f(x) ) as ( x ) approaches ( a ) from the right side (the larger values). We write it as $\lim_{x \to a^+} f(x)$.

2. Steps to Find One-Sided Limits from a Graph

Here’s how to find one-sided limits by looking at a graph:

• Step 1: Find the Point of Interest (a)
  Look for the number ( a ) where you want to check the limit. This could be a point where the function is smooth, has a gap, or a sudden change.

• Step 2: Look at the Left Side
  For the left-hand limit, see what the graph does as ( x ) gets closer to ( a ) from the left. You want to find out what value ( f(x) ) is getting close to. Draw an imaginary vertical line just to the left of ( a ) and follow the graph to see what value ( y ) (or ( f(x) )) is approaching.

• Step 3: Look at the Right Side
  For the right-hand limit, check what the graph does as ( x ) approaches ( a ) from the right. Again, draw an imaginary vertical line just to the right of ( a ) and see what value ( f(x) ) is getting close to.

• Step 4: Compare the Limits
  If both the left-hand limit $\lim_{x \to a^-} f(x)$ and the right-hand limit $\lim_{x \to a^+} f(x)$ exist and are the same, then the two-sided limit exists. We can show it as $\lim_{x \to a} f(x) = L$, where ( L ) is that common value. If they are not the same, then the two-sided limit does not exist.

3. Examples

Let’s look at a simple piece of a function:

• If ( x < 1 ), then ( f(x) = 2x + 1 ).

• If ( x \geq 1 ), then ( f(x) = -x + 3 ).

• Check Limits at ( x = 1 ):

  • Left-hand limit:
    $\lim_{x \to 1^-} f(x) = 2(1) + 1 = 3$
  • Right-hand limit:
    $\lim_{x \to 1^+} f(x) = -1 + 3 = 2$

Since the left-hand limit (3) is not the same as the right-hand limit (2), the two-sided limit at ( x = 1 ) does not exist.

4. Conclusion

Finding one-sided limits from a graph means looking closely at how a function acts as it gets close to a specific point from different sides. Being able to visually understand these limits is a key skill in calculus and math.