1. **Problem 5: Evaluate why $\lim_{x \to 0} \frac{x}{|x|}$ does not exist.** The function is $f(x) = \frac{x}{|x|}$. We analyze the left-hand and right-hand limits: - For $x \to 0^+$, $|x| = x$, so $f(x) = \frac{x}{x} = 1$. - For $x \to 0^-$, $|x| = -x$, so $f(x) = \frac{x}{-x} = -1$. Since the left-hand limit is $-1$ and the right-hand limit is $1$, the two-sided limit does not exist because the limits from both sides are not equal. 2. **Problem 6: Explain why $\lim_{x \to 1} \frac{1}{x-1}$ does not exist.** Consider $f(x) = \frac{1}{x-1}$. - As $x \to 1^+$, $x-1$ is a small positive number, so $f(x) \to +\infty$. - As $x \to 1^-$, $x-1$ is a small negative number, so $f(x) \to -\infty$. The left and right limits approach different infinities, so the limit does not exist. 3. **Problem 7: Can anything be said about $\lim_{x \to c} f(x)$ if $f$ is undefined at $x=c$?** The existence of $\lim_{x \to c} f(x)$ depends on the behavior of $f(x)$ near $c$, not on $f(c)$ itself. A limit can exist even if $f(c)$ is undefined, as limits depend on values arbitrarily close to $c$. 4. **Problem 8: Can anything be said about $\lim_{x \to 0} f(x)$ if $f$ is defined on $[-1,1]$?** Being defined on $[-1,1]$ means $f$ has values near 0, but the limit at 0 exists only if the left and right limits at 0 are equal. So, no guarantee the limit exists without more information. 5. **Problem 9: If $\lim_{x \to 1} f(x) = 5$, must $f$ be defined at $x=1$? Must $f(1) = 5$?** No, $f$ need not be defined at $x=1$ for the limit to exist. Also, even if $f(1)$ is defined, it need not equal 5. The limit concerns values near 1, not necessarily the value at 1. 6. **Problem 10: If $f(1) = 5$, must $\lim_{x \to 1} f(x)$ exist? If it does, must it equal 5?** No, $f(1) = 5$ does not guarantee the limit exists. If the limit exists, it does not have to equal 5. The limit depends on behavior near 1, not just the function value at 1. **Summary:** Limits depend on the behavior of $f(x)$ near the point, not necessarily on the function's value at that point. Limits may fail to exist if left and right limits differ or if the function grows without bound near the point.