This problem tests defining limits and using limit notation for functions with known limit properties like sin(x)/x. The expression \(\lim_{x \to 0} u(x) = 1\) correctly states the well-known limit of sin(x)/x approaching 1 as x nears 0. Despite u(0)=3, the notation focuses on values around x=0, not at it. This exemplifies standard limits overriding point definitions. A tempting distractor like \(\lim_{x \to 0} u(x) = 3\) fails because it confuses the redefined function value with the actual limit. For limit notation, recall standard limits, specify approach to the point, equate to the known value, and separate from f(a).

This problem tests defining limits and using limit notation for algebraic functions with potential discontinuities. The expression $\lim_{x \to 3} h(x) = 6$ correctly describes the behavior as x approaches 3, since simplifying $\frac{x^2-9}{x-3}$ yields $x+3$, which nears 6. Even though $h(3)=10$ is defined differently, the limit notation captures the nearby values, not the point itself. This illustrates removable discontinuities where limits exist despite redefinition. A tempting choice like $\lim_{x \to 3} h(x) = 10$ fails because it incorrectly uses the function value at x=3 instead of the approached value. Remember in limit notation to specify the variable and approach point, equate to the value from surrounding points, simplify expressions if needed, and separate from f(a).

This question examines defining limits and using limit notation for piecewise-defined functions. The expression \(\lim_{x \to 0} r(x) = 1\) is correct because from the left, x+1 approaches 1, and from the right, 1-x approaches 1 as x nears 0. Although r(0)=2, the limit notation describes the agreement in approaching 1, not the value at 0. This demonstrates limits at points of redefinition in piecewise functions. A misleading choice like \(\lim_{x \to 0} r(x) = 2\) fails as it uses r(0) instead of the nearby behavior. For limit notation, ensure left and right limits match, specify the approach point, equate to that common value, and differentiate from f(a).

This question tests the skill of defining limits and using limit notation. The expression in choice B, $\lim_{x\to1} k(x)=3$, is valid because simplifying gives $2x+1$ approaching 3 as $x$ approaches 1. Factoring removes the discontinuity. Both sides approach 3. A tempting distractor is choice C, $k(1)=3$, but it's undefined there. Use $\to$ and simplify for removable discontinuities.

This question tests the skill of defining limits and using limit notation. The expression in choice C, $\lim_{x\to1} h(x)=1$, is valid because the limit of $\ln(x)/(x-1)$ as $x$ approaches 1 is 1, a standard form. It's $0/0$ indeterminate, resolved by L'Hôpital to $1/x$ over 1, approaching 1. Both sides agree. A tempting distractor is choice A, $\lim_{x\to1} h(x)=0$, from numerator but ignoring derivative. Use $\to$ and apply rules for indeterminates.

This question tests limit notation for piecewise functions. The limit as x approaches 0 of t(x) is 3 because the left-hand limit from x+3 approaches 3 and the right-hand limit from 3-x also approaches 3. The function value at x=0 is 3 from the left piece, but the limit is independent of this. Therefore, the notation \(\lim_{x\to0} t(x)=3\) accurately represents the behavior. A tempting distractor is B, which states t(0)=0, perhaps from misreading the piecewise definition. A checklist for limit notation includes specifying the function, the variable approaching a value, checking one-sided limits agree, and stating the limit value or that it does not exist.

This question tests limit notation for trigonometric limits. The limit as x approaches 0 of s(x) is 3 because the expression simplifies to 3 times sin(3x)/(3x), which approaches 3*1=3 using the standard limit of sin(u)/u as u approaches 0. The function value s(0)=0 does not affect the limit, which examines values near 0. Therefore, the notation \(\lim_{x\to0} s(x)=3\) is correct. A tempting distractor is A, which mistakenly uses the function value at x=0 for the limit. A checklist for limit notation includes specifying the function, the variable approaching a value, checking one-sided limits agree, and stating the limit value or that it does not exist.

This question tests limit notation when one-sided limits differ. The limit as x approaches 5 of q(x) does not exist because the left-hand limit is 1 and the right-hand limit is 4, which do not match. The value of q(5), if defined, is irrelevant to whether the overall limit exists. Therefore, the expression \(\lim_{x\to5} q(x)\) does not exist correctly describes the situation. A tempting distractor is C, which might incorrectly take the limit as one of the one-sided values. A checklist for limit notation includes specifying the function, the variable approaching a value, checking one-sided limits agree, and stating the limit value or that it does not exist.

This question tests the skill of defining limits and using limit notation. The expression $\lim_{x\to0}$ n(x)=\frac{1}{2}$ is correct because rationalizing yields ($\sqrt{x+1}$-1)/x = 1/($\sqrt{x+1}$+1) → 1/2. Both sides approach 1/2. This notation equals the limit value. A tempting distractor is C, n(0)=\frac{1}{2}$, but n(0) undefined. To check limit notation, rationalize expressions, use →, and compute the value.

This question tests the skill of defining limits and using limit notation. The expression in choice C, $\lim_{x\to2} f(x)$ does not exist, is valid because the left limit is 6 and the right is 5, differing. Left uses $3x$ approaching 6, right $x^2+1$ approaching 5. Mismatch means no two-sided limit. A tempting distractor is choice A, $\lim_{x\to2} f(x)=7$, confusing with the point value. Use $\to$ and evaluate one-sided for piecewise.