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.

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.

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.

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.

This piecewise function has t(x) = x² for x < 0 and t(x) = 3 for x > 0. The question asks specifically about right-hand behavior as x → 0. From the right (x > 0), we use t(x) = 3, so $lim_{x→0^+}$ t(x) = 3. From the left (x < 0), we use t(x) = x², so $lim_{x→0^-}$ t(x) = 0² = 0. The correct notation for the right-hand limit is $lim_{x→0^+}$ t(x) = 3, properly indicating approach from the positive side. Option E incorrectly states the right-hand limit is 0, which would be the left-hand limit. A common error is confusing which formula applies for each direction. Limit notation checklist: x → 0^+ means x > 0 (approaching from the right), x → 0^- means x < 0 (approaching from the left), and match the correct piece of the function to each direction.

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.

The function $u(x) = \frac{(x-5)(x+1)}{x-5}$ simplifies to $u(x) = x+1$ for all $x \neq 5$ by canceling the common factor $(x-5)$. As $x$ approaches 5, the simplified function approaches $5+1=6$, regardless of the assigned value $u(5)=-3$. The correct limit notation is $\lim_{x\to 5} u(x) = 6$. This illustrates a key principle: removable discontinuities don't affect limits—the limit exists even though the original expression is undefined at $x=5$. A common error is thinking the limit must equal the assigned function value, or writing $\lim_{x\to 5} u(5)$, which incorrectly evaluates the function inside the limit. Notation checklist: (1) $\lim$ symbol, (2) $x\to 5$, (3) $u(x)$ without substitution, (4) equals 6.

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.

The table shows q(x) values approaching 6 as x approaches 3 from both sides: from the left (2.9 → 5.98, 2.99 → 5.998) and from the right (3.01 → 6.002, 3.1 → 6.02). Since the values approach 6 from both directions, the two-sided limit exists and equals 6. The correct notation is $lim_{x→3}$ q(x) = 6, using the standard two-sided limit notation without directional superscripts. Option A incorrectly states q(3) = 6, which is a function value, not a limit statement. A common error is using overly specific decimal values (like 5.98 or 6.02) instead of recognizing the pattern approaching 6. Limit notation checklist: use $lim_{x→a}$ for two-sided limits, omit superscripts when approaching from both sides, and identify the limiting value from the pattern in the table.

The function r(x) = (√x - 2)/(x - 4) is undefined at x = 4, but we can find the limit by rationalizing. Multiplying by (√x + 2)/(√x + 2), we get r(x) = (x - 4)/[(x - 4)(√x + 2)] = 1/(√x + 2) for x ≠ 4. As x approaches 4, r(x) approaches 1/(√4 + 2) = 1/(2 + 2) = 1/4. The correct notation is $lim_{x→4}$ r(x) = 1/4, using the two-sided limit since the simplified form approaches the same value from both directions. Option E incorrectly uses x = 4 instead of x → 4, which is improper limit notation. A common error is forgetting to simplify before evaluating the limit. Limit notation checklist: use → not = in limits, simplify indeterminate forms before evaluating, and use two-sided notation when both one-sided limits agree.