The function $s(x) = \frac{|x|}{x}$ equals $\frac{x}{x} = 1$ when $x>0$ and $\frac{-x}{x} = -1$ when $x<0$. For the right-hand limit as $x\to 0^+$, we consider positive values of $x$ approaching 0, where $s(x) = 1$. Therefore, $\lim_{x\to 0^+} s(x) = 1$ is the correct notation. The assigned value $s(0)=0$ is irrelevant to the limit calculation—limits describe behavior near a point, not at it. A common error is writing $\lim_{x\to 0^+} s(0)$, which incorrectly substitutes 0 into the function within the limit notation. Note that the left-hand limit would be -1, so the two-sided limit doesn't exist. Notation checklist: (1) $\lim$ symbol, (2) $x\to 0^+$ for right approach, (3) $s(x)$ not $s(0)$, (4) equals 1.

Limit notation is used to describe the value a function approaches as the input nears a certain point, distinct from the function's value at that point. In this scenario, the table indicates that f(x) approaches 5 as x gets close to 2 from both sides, making $\lim_{x\to 2}$ f(x)=5 the appropriate expression. This notation is valid because it focuses on the behavior around x=2 without evaluating f at exactly 2. A common symbolic error is writing $\lim_{x\to 2}$ f(2)=5, which improperly substitutes the point into the function inside the limit. Another frequent mistake is using f(2)=5, which represents the function value, not the limit. Remember, limits can exist even if the function is undefined at the point. Transferable notation checklist: 1. Use 'lim' to denote limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Limit notation conveys approaching values, as the table shows t(x) nearing 6 as x approaches 1. Hence, $\lim_{x\to 1} t(x)=6$ is appropriate for this behavior. This is valid regardless of $t(1)$. An error is $\lim_{x\to 1} t(1)=6$, misplacing evaluation. $t(1)=6$ is function value, not limit. Distinguish clearly. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with $x \to a$. 3. Add $^{+}$ or $^{-}$ for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Limit notation describes functional behavior near a point, with the table indicating q(x) approaches 0 as x nears -2. Thus, $\lim_{x\to -2}$ q(x)=0 represents this two-sided approach. This is valid as it ignores q(-2) and focuses on vicinity. A common error is $\lim_{x\to -2}$ q(-2)=0, blending notation improperly. q(-2)=0 denotes value at -2, not limit. Limits exist independently of point values. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

Table suggests c(x) to -1 at x=4. $\lim_{x\to 4}$ c(x)=-1 represents. Valid both sides. Error: $\lim_{x\to 4}$ c(4)=-1. c(4)=-1 value. Separate. Transferable notation checklist: 1. Use 'lim' for limits. 2. Specify the approach with x\to a. 3. Add ^+ or ^- for one-sided limits if needed. 4. Ensure the expression equals the approached value.

If left and right limits differ, the two-sided limit does not exist, denoted as DNE in notation. For s(x) = |x|/x, it approaches 1 from the right and -1 from the left as x nears 0, so $\lim_{x \to 0}$ s(x) = DNE is accurate. This is valid when sides disagree, despite s(0) = 0. A common error is claiming a limit value like 0, confusing with the function at 0. Another symbolic mistake is reversing, such as $\lim_{s(x) \to 0}$ x = 1, or using one-sided without specifying DNE for two-sided. Always check both directions for existence. Transferable notation checklist: 1. Write as $\lim_{x \to a}$ f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

One-sided limit notation specifies direction using + for right or - for left, which is crucial when behavior differs on each side. The table shows h(x) approaching -2 only from the right as x nears 0, so $\lim_{x \to 0^+}$ h(x) = -2 accurately represents this. This notation is valid as it matches the directional approach described. A common symbolic error is omitting the direction, like $\lim_{x \to 0}$ h(x) = -2, assuming two-sided without evidence. Another mistake is confusing the limit with function equality, such as $\lim_{x \to 0}$ h(x) = h(0). Always check if the data specifies one side or both. Transferable notation checklist: 1. Write as $\lim_{x \to a}$ f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

Limit notation for rational functions often involves simplifying expressions to find the approaching value, ignoring the function's defined value at the point. For g(x) = $(x^2$ - 9)/(x - 3) simplified to x + 3 for x ≠ 3, the limit as x approaches 3 is 6, despite g(3) = 10, so $\lim_{x \to 3}$ g(x) = 6 is correct. This is valid because limits consider behavior near the point, not at it. A common error is using the redefined value, like $\lim_{x \to 3}$ g(x) = 10. Another mistake is reversing the limit, such as $\lim_{x \to 10}$ g(x) = 3 or $\lim_{g(x) \to 3}$ x = 6. The two-sided limit applies here as the function approaches the same value from both sides. Transferable notation checklist: 1. Write as $\lim_{x \to a}$ f(x) = L, with x approaching a and f(x) to L. 2. Use + or - for one-sided limits if specified. 3. Do not equate limit to f(a) unless continuous. 4. Avoid swapping x and f(x) roles. 5. Confirm left and right agreement for two-sided limits.

The function $g$ has a jump discontinuity at $x=3$: it equals 1 for $x<3$, jumps to 5 at $x=3$, then equals 4 for $x>3$. When approaching from the right (values greater than 3), we're in the region where $g(x)=4$, so the right-hand limit is $\lim_{x\to 3^+} g(x) = 4$. The superscript plus sign indicates we only consider values approaching from the right. A common mistake is thinking the limit must involve the function value at the point—but $g(3)=5$ is irrelevant to the right-hand limit. Another error is writing $\lim_{x\to 3} g(3)$, which incorrectly substitutes the value into the limit notation. Notation checklist: (1) $\lim$ symbol, (2) $x\to 3^+$ for right-hand approach, (3) $g(x)$ not $g(3)$, (4) equals 4.

The function $r$ is defined as $r(x)=2$ for $x\leq 4$ and $r(x)=2+(x-4)$ for $x>4$. For the right-hand limit as $x\to 4^+$, we use the formula for $x>4$: as $x$ approaches 4 from the right, $(x-4)$ approaches 0, so $r(x)$ approaches $2+0=2$. The correct notation is $\lim_{x\to 4^+} r(x) = 2$, where the superscript plus indicates we only consider values greater than 4. Notice that both one-sided limits equal 2, making the function continuous at $x=4$. A common error is writing $\lim_{x\to 4^+} r(4)$, which incorrectly evaluates the function at 4 rather than describing the limiting behavior. Notation checklist: (1) $\lim$ symbol, (2) $x\to 4^+$ for right approach, (3) $r(x)$ not $r(4)$, (4) equals 2.