For f(x) = 3/(x-2), we need to identify the limit notation for the behavior as x approaches 2 from the right. The notation x→2⁺ means x approaches 2 from values greater than 2. As x gets closer to 2 from the right, the denominator (x-2) becomes a small positive number, making the fraction 3/(small positive) = large positive value. Therefore, lim[x→2⁺] f(x) = +∞ correctly represents this behavior. A common error is using an equals sign instead of an arrow (x=2⁺), which is incorrect notation. Another mistake is evaluating f(2), which is undefined since division by zero is not allowed. Notation checklist: Use → not =, include superscript + or - for one-sided limits, and remember that limits describe approach behavior, not function values.

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.

The coefficient's sign affects the infinity direction in rational functions. For $g(x) = \dfrac{-4}{x+6}$, as $x \to -6^-$, $x+6$ negative small, $-4$/negative = positive large, so $\lim_{x\to -6^-} g(x) = +\infty$ correctly represents this. This notation is valid due to the negative numerator over negative denominator yielding positive. A common error is ignoring the numerator's sign, leading to incorrect $-\infty$. Using two-sided limit here would be wrong since sides differ. 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 ($+\infty$ or $-\infty$) 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.

Infinite limits are denoted when a function approaches positive or negative infinity near a point, often indicating a vertical asymptote. For g(x) = x²/(x²-1), as x approaches 1 from the left, the denominator approaches 0 from the negative side while the numerator is positive, resulting in g(x) approaching $-\infty$, so $\lim_{x\to 1^-} g(x) = -\infty$ is correct. This representation is valid because near x=1^-, (x-1) is negative and (x+1) positive, making the denominator negative, and positive over negative yields negative values growing in magnitude. A common symbolic error is writing g(1) = -∞, but limits do not assign a value to the function at the point. Confusing left and right directions is another frequent mistake, which changes the sign analysis. 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 ($+\infty$ or $-\infty$) 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 $g(x) = \dfrac{3}{x+1}$, the vertical asymptote is at $x=-1$ where the denominator is zero. As $x$ approaches -1 from the left ($x < -1$), $x+1$ is negative and approaching zero, so $g(x) = \dfrac{3}{\text{small negative}}$ becomes a large negative number, hence $-\infty$. The correct notation is $\lim_{x \to -1^-} g(x) = -\infty$, emphasizing the left-hand approach and negative infinity. This representation is valid as it describes the unbounded decrease without evaluating at $x=-1$. A common error is using two-sided limit notation when sides differ, or mistakenly writing $g(-1) = -\infty$, but limits describe approach, not point values. Students often mix up signs by not carefully checking the denominator's sign from each side. Transferable notation checklist: Identify if the limit is one-sided or two-sided; determine the sign of the expression near the asymptote; use $\pm \infty$ appropriately; avoid equating to function value; specify direction with $^{+}$ or $^{-}$ when necessary.

Direction matters in determining the sign for limits near asymptotes. For g(x) = -4/(x+6), as $x \to -6^+$, $x+6$ positive small, $-4/\text{positive} = \text{negative large}$, so $\lim_{x\to -6^+} g(x) = -\infty$ is the correct notation. This is valid because the negative numerator over positive denominator results in large negative values. A common symbolic error is misplacing the direction superscript, changing the behavior described. Writing $g(-6) = -\infty$ confuses limit with function value. 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 f(x) = 2/(x(x-4)), as x→4^-, x positive ≈4, (x-4) negative small, denominator negative small, 2/negative = -∞. The notation $lim_{x→4^-}$ f(x) = -∞ fits. This is valid based on left-hand signs. A frequent mistake is using +∞ or equating to f(4) = -∞. Failing to check the (x-4) negativity 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.

For rational functions, sign analysis near poles determines the infinity direction. For f(x) = 1/(x-8), as x → 8^-, the denominator is negative small, so f(x) → -∞, and \(\lim_{x\to 8^-} f(x) = -\infty\) is correct. This representation is valid based on the sign of the denominator approaching from the left. A common error is writing the two-sided limit when sides differ. Using f(8) = -∞ is symbolically wrong, as it's not a function value. 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 p(x) = $1/(x-3)^2$, the denominator is always positive near x=3 except at the point itself, leading to +∞ from both sides. As x approaches 3 from either side, $(x-3)^2$ is positive small, so p(x) becomes large positive. The two-sided limit notation $lim_{x→3}$ p(x) = +∞ is appropriate since both one-sided limits agree. This is valid for even-powered denominators causing symmetric behavior to +∞. A common error is writing one-sided notation unnecessarily or using p(3) = +∞, but limits aren't function values. Students sometimes incorrectly predict -∞ by ignoring the squaring effect. 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 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.