Enter xx-values into the table below to determine function values for a function f, of, xf(x) for various inputs. Use the function values in order to determine the answers to each question below.
In the table below, enter a number in the input box in the x column and then press the "try value" button in the f of x column. After you press the button, your input and the corresponding output will be placed on the table and a new input box will be made available. After the table, a "clear table" button allows you to remove all the rows you've created.
xx	f, of, xf(x)
Answer
Attempt 1 out of 2

limit, start subscript, x, right arrow, 1, start superscript, minus, end superscript, end subscript, f, of, x, equals
x→1
−

lim
​
f(x)=
limit, start subscript, x, right arrow, 1, start superscript, plus, end superscript, end subscript, f, of, x, equals
x→1
+

lim
​
f(x)=
limit, start subscript, x, right arrow, 1, end subscript, f, of, x, equals
x→1
lim
​
f(x)=
f, of, 1, equalsf(1)=


To solve the problem, you need to determine the function's behavior as \( x \) approaches 1 from the left (\( x \to 1^- \)), from the right (\( x \to 1^+ \)), and exactly at \( x = 1 \). To do this, you should follow these steps:

### Steps to Determine the Limits:

1. **Enter Values Near 1:**
   - Input values slightly less than 1 (like 0.9, 0.99, 0.999) to find the left-hand limit (\( \lim_{x \to 1^-} f(x) \)).
   - Input values slightly greater than 1 (like 1.1, 1.01, 1.001) to find the right-hand limit (\( \lim_{x \to 1^+} f(x) \)).

2. **Check the Limit as \( x \) Approaches 1:**
   - Observe the behavior of \( f(x) \) as \( x \) gets closer to 1 from both sides. This will help determine the overall limit \( \lim_{x \to 1} f(x) \).

3. **Evaluate the Function at \( x = 1 \):**
   - Calculate \( f(1) \) to see the function's value at \( x = 1 \).

### Example Approach:

- **For \( x \to 1^- \):** Try values like 0.9, 0.99, 0.999.
- **For \( x \to 1^+ \):** Try values like 1.1, 1.01, 1.001.
- **For \( x = 1 \):** Evaluate \( f(1) \) directly.

### Conclusion:
1. \(\lim_{x \to 1^-} f(x) = \) [insert value]
2. \(\lim_{x \to 1^+} f(x) = \) [insert value]
3. \(\lim_{x \to 1} f(x) = \) [insert value, if both left and right limits match]
4. \(f(1) = \) [insert value]

This will give you the answers you need for each limit and the value at \( x = 1 \).

If you input the values in an actual function table, please provide the outputs for further assistance. Otherwise, let me know if there’s a specific function or table you’re working with, and I can guide you more accurately.

Would you like more detailed instructions on solving limits?

### 5 Related Questions
1. What happens if the left-hand and right-hand limits are different?
2. How do you handle a discontinuity at a point like \( x = 1 \)?
3. What are some common types of discontinuities in functions?
4. Can a function have a limit at \( x = 1 \) if \( f(1) \) is undefined?
5. What is the difference between one-sided and two-sided limits?

### Tip
Always compare the left-hand limit and the right-hand limit to determine if the overall limit exists at a given point.

Learn how to solve problems involving limits in calculus. This page provides detailed instructions on determining left-hand limits, right-hand limits, and limits at specific points. Explore examples and step-by-step solutions for evaluating functions near a point.

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