For f(x) = $2/(x+5)^2$, the even power ensures positive denominator near x=-5 from both sides, leading to +∞. The two-sided limit $lim_{x→-5}$ f(x) = +∞ is correct. This is valid for symmetric positive behavior. A common error is predicting differing sides or using f(-5) = +∞. Overlooking the square's effect is typical. Transferable notation checklist: Identify if the limit is one-sided or two-sided; determine the sign of the expression near the asymptote; use ±∞ appropriately; avoid equating to function value; specify direction with ^+ or ^- when necessary.

For r(x) = 4/(x-3)³, we analyze the behavior as x approaches 3 from the left (x→3⁻). When x is slightly less than 3, the denominator (x-3) is a small negative number. Since we're cubing this negative value, (x-3)³ remains negative. The fraction becomes 4/(small negative) = large negative value. Therefore, lim[x→3⁻] r(x) = -∞ correctly represents this behavior. A common error is forgetting that odd powers preserve the sign of negative numbers, unlike even powers. Another mistake is writing r(3) = -∞, which incorrectly suggests the function has a value at x = 3. Notation checklist: Pay attention to the power of the denominator, verify signs carefully for odd vs even powers, and use proper limit notation.

For u(x) = 6/(x+4)², we examine the behavior as x approaches -4. Since the denominator is squared, (x+4)² is always positive for x ≠ -4. As x approaches -4 from either direction, the denominator approaches 0 while remaining positive. The fraction becomes 6/(small positive) = large positive value. Since this happens from both sides, lim[x→-4] u(x) = +∞ correctly represents this behavior. A common error is thinking that squared denominators might lead to different one-sided limits, but squares are always non-negative. Another mistake is writing u(-4) = +∞ instead of using limit notation. Notation checklist: Remember that even powers create the same behavior from both sides, two-sided limits exist when both one-sided limits equal +∞, and always use limit notation for infinite values.

For g(x) = -5/(x+1), we analyze the behavior as x approaches -1 from the left. The notation x→-1⁻ means x approaches -1 from values less than -1. When x is slightly less than -1, the denominator (x+1) is a small negative number. The fraction becomes -5/(small negative) = -5/(-small) = large positive value. Therefore, lim[x→-1⁻] g(x) = +∞ correctly represents this behavior. A common error is confusing the direction of approach or the sign of infinity. Another mistake is writing g(-1) = +∞, which incorrectly suggests the function has a value at x = -1. Notation checklist: Verify the sign of the numerator and denominator separately, use proper limit notation with arrows, and distinguish between left (-) and right (+) approach.

For h(x) = 1/(x(x-4)), as x→0^-, x negative small, (x-4) negative, denominator positive small (negative*negative), 1/positive = +∞. The notation $lim_{x→0^-}$ h(x) = +∞ matches. This is valid via sign analysis. A symbolic mistake is predicting -∞ or using h(0) = +∞. Confusing sides is common. Transferable notation checklist: Identify if the limit is one-sided or two-sided; determine the sign of the expression near the asymptote; use ±∞ appropriately; avoid equating to function value; specify direction with ^+ or ^- when necessary.

The function q(x) = (x+4)/(x+4) simplifies to 1 for x ≠ -4, so there's a removable discontinuity at x=-4, but the limit exists. As x approaches -4 from both sides, q(x) approaches 1. The correct notation is $lim_{x→-4}$ q(x) = 1, indicating a finite two-sided limit. This is valid because after simplification, the function is constant near x=-4. A common symbolic error is failing to simplify and assuming infinite limit like +∞, or writing q(-4) = 1 when it's undefined. Mixing it with asymptotic behavior without checking is frequent. Transferable notation checklist: Identify if the limit is one-sided or two-sided; determine the sign of the expression near the asymptote; use ±∞ appropriately; avoid equating to function value; specify direction with ^+ or ^- when necessary.

Limit notation specifies direction and infinity type to describe asymptotic behavior accurately. For f(x) = 1/(x²-1), as x approaches -1 from the right, the denominator approaches 0 from the negative side, so f(x) approaches -∞, making \(\lim_{x\to -1^+} f(x) = -\infty\) the correct choice. This is valid since (x+1)>0 small and (x-1)<0, product negative, reciprocal large negative. A common symbolic error is omitting the direction superscript, implying a two-sided limit that doesn't exist here. Misjudging the sign by not evaluating factor signs is another frequent issue. To ensure correct notation, use this transferable notation checklist: 1. Use 'lim' to indicate a limit, not the function equals notation. 2. Specify the direction with ^+ or ^- for one-sided limits. 3. Determine the sign (+∞ or -∞) based on whether the function increases or decreases without bound. 4. Remember that vertical asymptotes correspond to limits to infinity, and the function is undefined at that point.

The function u(x) = $-1/(x-1)^2$ has an even-powered denominator, always positive near x=1, with negative numerator leading to -∞ from both sides. As x approaches 1, $(x-1)^2$ positive small, u(x) = -1/(positive) = large negative. The two-sided notation $lim_{x→1}$ u(x) = -∞ is correct since sides match. This is valid for symmetric negative infinite behavior. A common symbolic error is predicting +∞ by overlooking the negative sign, or writing u(1) = -∞, but limits aren't point evaluations. Ignoring the even power's positivity is frequent. Transferable notation checklist: Identify if the limit is one-sided or two-sided; determine the sign of the expression near the asymptote; use ±∞ appropriately; avoid equating to function value; specify direction with ^+ or ^- when necessary.

Sign analysis for k(x) = 2x/(x²-16) near x=4 involves checking numerator and denominator. As x → 4^+, denominator positive small, numerator positive, so k(x) → +∞, and \(\lim_{x\to 4^+} k(x) = +\infty\) is correct. This notation is valid because (x-4)>0, (x+4)>0, overall positive. A common symbolic error is incorrect sign from misfactoring. Using wrong direction like -4^+ is a distraction. To ensure correct notation, use this transferable notation checklist: 1. Use 'lim' to indicate a limit, not the function equals notation. 2. Specify the direction with ^+ or ^- for one-sided limits. 3. Determine the sign (+∞ or -∞) based on whether the function increases or decreases without bound. 4. Remember that vertical asymptotes correspond to limits to infinity, and the function is undefined at that point.

For k(x) = 2x/(x²-16), as x → 4^-, (x-4) negative, (x+4) positive, denominator negative, numerator positive, so k(x) → -∞, making \(\lim_{x\to 4^-} k(x) = -\infty\) the valid choice. This represents the left-side behavior accurately. The sign is determined by the negative denominator dominating. A common error is assuming +∞ without sign check. Writing the two-sided limit as 0 ignores the asymptote. To ensure correct notation, use this transferable notation checklist: 1. Use 'lim' to indicate a limit, not the function equals notation. 2. Specify the direction with ^+ or ^- for one-sided limits. 3. Determine the sign (+∞ or -∞) based on whether the function increases or decreases without bound. 4. Remember that vertical asymptotes correspond to limits to infinity, and the function is undefined at that point.