1. **Problem Statement:** Explain why the limits do not exist for the following: 5. $$\lim_{x \to 0} |x|$$ 6. $$\lim_{x \to 1} \frac{1}{x - 1}$$ 2. **Limit Definition and Important Rules:** The limit $$\lim_{x \to a} f(x)$$ exists if and only if the left-hand limit $$\lim_{x \to a^-} f(x)$$ and the right-hand limit $$\lim_{x \to a^+} f(x)$$ both exist and are equal. 3. **Exercise 5: $$\lim_{x \to 0} |x|$$** - The function $$|x|$$ is defined as: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ - Evaluate the left-hand limit: $$\lim_{x \to 0^-} |x| = \lim_{x \to 0^-} -x = 0$$ - Evaluate the right-hand limit: $$\lim_{x \to 0^+} |x| = \lim_{x \to 0^+} x = 0$$ - Since both one-sided limits equal 0, the limit exists and equals 0. - **Conclusion:** The limit $$\lim_{x \to 0} |x|$$ exists and equals 0, so the limit does exist here. 4. **Exercise 6: $$\lim_{x \to 1} \frac{1}{x - 1}$$** - Evaluate the left-hand limit: As $$x \to 1^-$$, $$x - 1$$ approaches 0 from the negative side, so $$\frac{1}{x - 1} \to -\infty$$. - Evaluate the right-hand limit: As $$x \to 1^+$$, $$x - 1$$ approaches 0 from the positive side, so $$\frac{1}{x - 1} \to +\infty$$. - Since the left-hand and right-hand limits are not equal (one tends to $$-\infty$$, the other to $$+\infty$$), the limit does not exist. 5. **Exercise 7:** - If $$f(x)$$ is defined for all real $$x$$ except at $$x = c$$, nothing definitive can be said about $$\lim_{x \to c} f(x)$$ without more information. - The limit may exist if the left and right limits at $$c$$ exist and are equal. - The function's value at $$c$$ is irrelevant to the existence of the limit. 6. **Exercise 8:** - If $$f(x)$$ is defined for all $$x \in [-1,1]$$, the limit $$\lim_{x \to 0} f(x)$$ may or may not exist. - Existence depends on the behavior of $$f(x)$$ near 0. - Being defined on the interval does not guarantee the limit exists. 7. **Exercise 9:** - If $$\lim_{x \to 1} f(x) = 5$$, $$f$$ need not be defined at $$x=1$$. - If $$f(1)$$ is defined, it does not have to equal 5. - The limit describes the behavior near 1, not necessarily the function value at 1. 8. **Exercise 10:** - If $$f(1) = 5$$, the limit $$\lim_{x \to 1} f(x)$$ may or may not exist. - If the limit exists, it does not have to equal 5. - The function value at a point does not guarantee the limit at that point. **Final answers:** - For Exercise 5, the limit exists and equals 0. - For Exercise 6, the limit does not exist because the left and right limits are not equal. - For Exercises 7 to 10, the existence and value of limits depend on the function's behavior near the point, not just on the function's definition or value at that point.