15. $$ \lim _{x \rightarrow 4} \frac{v^{2}-6 v+8}{2 v^{2}-8 v} $$ 16. $$ \lim _{x \rightarrow 3} \frac{x^{2}-8 x+15}{x^{2}-7 x+12} $$ 17. $$ \lim _{h \rightarrow \frac{1}{2}} \frac{4 h^{2}+4 h-3}{2 h-1} $$ 18. $$ \lim _{x \rightarrow \frac{2}{3}} \frac{3 x-2}{3 x^{2}-11 x+6} $$ 19. $$ \lim _{w \rightarrow-\frac{4}{3}} \frac{9 w^{2}+9 w-4}{3 w^{2}+7 w+4} $$ 20. $$ \lim _{y \rightarrow 6} \frac{2 y^{2}-15 y+18}{3 y^{2}-17 y-6} $$ 21. $$ \lim _{x \rightarrow 5} \frac{2 x^{2}-13 x+15}{x^{2}-x-20} $$ 22. $$ \lim _{x \rightarrow-\frac{1}{3}} \frac{9 x^{2}-1}{6 x^{2}+5 x+1} $$ 23. $$ \lim _{y \rightarrow-1} \frac{y+1}{y^{3}+1} $$

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Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{yightarrow -1}\left(\frac{y+1}{y^{3}+1}ight)$$
- step1: Reduce the fraction:
  $$\lim _{yightarrow -1}\left(\frac{1}{y^{2}-y+1}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{yightarrow -1}\left(1ight)}{\lim _{yightarrow -1}\left(y^{2}-y+1ight)}$$
- step3: Evaluate:
  $$\frac{1}{\lim _{yightarrow -1}\left(y^{2}-y+1ight)}$$
- step4: Evaluate:
  $$\frac{1}{3}$$
Calculate the limit $$ \lim_{x ightarrow \frac{2}{3}} \frac{3 x-2}{3 x^{2}-11 x+6} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{xightarrow \frac{2}{3}}\left(\frac{3x-2}{3x^{2}-11x+6}ight)$$
- step1: Reduce the fraction:
  $$\lim _{xightarrow \frac{2}{3}}\left(\frac{1}{x-3}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{xightarrow \frac{2}{3}}\left(1ight)}{\lim _{xightarrow \frac{2}{3}}\left(x-3ight)}$$
- step3: Evaluate:
  $$\frac{1}{\lim _{xightarrow \frac{2}{3}}\left(x-3ight)}$$
- step4: Evaluate:
  $$\frac{1}{-\frac{7}{3}}$$
- step5: Multiply by the reciprocal:
  $$-\frac{3}{7}$$
Calculate the limit $$ \lim_{v ightarrow 4} \frac{v^{2}-6 v+8}{2 v^{2}-8 v} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{vightarrow 4}\left(\frac{v^{2}-6v+8}{2v^{2}-8v}ight)$$
- step1: Reduce the fraction:
  $$\lim _{vightarrow 4}\left(\frac{v-2}{v	imes 2}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{vightarrow 4}\left(v-2ight)}{\lim _{vightarrow 4}\left(v	imes 2ight)}$$
- step3: Evaluate:
  $$\frac{2}{\lim _{vightarrow 4}\left(v	imes 2ight)}$$
- step4: Evaluate:
  $$\frac{2}{8}$$
- step5: Reduce the fraction:
  $$\frac{1}{4}$$
Calculate the limit $$ \lim_{x ightarrow 3} \frac{x^{2}-8 x+15}{x^{2}-7 x+12} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{xightarrow 3}\left(\frac{x^{2}-8x+15}{x^{2}-7x+12}ight)$$
- step1: Reduce the fraction:
  $$\lim _{xightarrow 3}\left(\frac{x-5}{x-4}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{xightarrow 3}\left(x-5ight)}{\lim _{xightarrow 3}\left(x-4ight)}$$
- step3: Evaluate:
  $$\frac{-2}{\lim _{xightarrow 3}\left(x-4ight)}$$
- step4: Evaluate:
  $$\frac{-2}{-1}$$
- step5: Divide the terms:
  $$2$$
Calculate the limit $$ \lim_{y ightarrow 6} \frac{2 y^{2}-15 y+18}{3 y^{2}-17 y-6} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{yightarrow 6}\left(\frac{2y^{2}-15y+18}{3y^{2}-17y-6}ight)$$
- step1: Reduce the fraction:
  $$\lim _{yightarrow 6}\left(\frac{2y-3}{3y+1}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{yightarrow 6}\left(2y-3ight)}{\lim _{yightarrow 6}\left(3y+1ight)}$$
- step3: Evaluate:
  $$\frac{9}{\lim _{yightarrow 6}\left(3y+1ight)}$$
- step4: Evaluate:
  $$\frac{9}{19}$$
Calculate the limit $$ \lim_{w ightarrow -\frac{4}{3}} \frac{9 w^{2}+9 w-4}{3 w^{2}+7 w+4} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{wightarrow -\frac{4}{3}}\left(\frac{9w^{2}+9w-4}{3w^{2}+7w+4}ight)$$
- step1: Reduce the fraction:
  $$\lim _{wightarrow -\frac{4}{3}}\left(\frac{3w-1}{w+1}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{wightarrow -\frac{4}{3}}\left(3w-1ight)}{\lim _{wightarrow -\frac{4}{3}}\left(w+1ight)}$$
- step3: Evaluate:
  $$\frac{-5}{\lim _{wightarrow -\frac{4}{3}}\left(w+1ight)}$$
- step4: Evaluate:
  $$\frac{-5}{-\frac{1}{3}}$$
- step5: Multiply by the reciprocal:
  $$-5\left(-3ight)$$
- step6: Use the rules for multiplication and division:
  $$5	imes 3$$
- step7: Multiply the numbers:
  $$15$$
Calculate the limit $$ \lim_{h ightarrow \frac{1}{2}} \frac{4 h^{2}+4 h-3}{2 h-1} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{hightarrow \frac{1}{2}}\left(\frac{4h^{2}+4h-3}{2h-1}ight)$$
- step1: Reduce the fraction:
  $$\lim _{hightarrow \frac{1}{2}}\left(2h+3ight)$$
- step2: Rewrite the expression:
  $$\lim _{hightarrow \frac{1}{2}}\left(2hight)+\lim _{hightarrow \frac{1}{2}}\left(3ight)$$
- step3: Calculate:
  $$1+3$$
- step4: Calculate:
  $$4$$
Calculate the limit $$ \lim_{x ightarrow 5} \frac{2 x^{2}-13 x+15}{x^{2}-x-20} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{xightarrow 5}\left(\frac{2x^{2}-13x+15}{x^{2}-x-20}ight)$$
- step1: Reduce the fraction:
  $$\lim _{xightarrow 5}\left(\frac{2x-3}{x+4}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{xightarrow 5}\left(2x-3ight)}{\lim _{xightarrow 5}\left(x+4ight)}$$
- step3: Evaluate:
  $$\frac{7}{\lim _{xightarrow 5}\left(x+4ight)}$$
- step4: Evaluate:
  $$\frac{7}{9}$$
Calculate the limit $$ \lim_{x ightarrow -\frac{1}{3}} \frac{9 x^{2}-1}{6 x^{2}+5 x+1} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{xightarrow -\frac{1}{3}}\left(\frac{9x^{2}-1}{6x^{2}+5x+1}ight)$$
- step1: Reduce the fraction:
  $$\lim _{xightarrow -\frac{1}{3}}\left(\frac{3x-1}{2x+1}ight)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{xightarrow -\frac{1}{3}}\left(3x-1ight)}{\lim _{xightarrow -\frac{1}{3}}\left(2x+1ight)}$$
- step3: Evaluate:
  $$\frac{-2}{\lim _{xightarrow -\frac{1}{3}}\left(2x+1ight)}$$
- step4: Evaluate:
  $$\frac{-2}{\frac{1}{3}}$$
- step5: Multiply by the reciprocal:
  $$-2	imes 3$$
- step6: Multiply the numbers:
  $$-6$$
Aquí están los límites calculados para cada una de las expresiones:

15. $$ \lim _{v ightarrow 4} \frac{v^{2}-6 v+8}{2 v^{2}-8 v} = \frac{1}{4} $$ o $$ 0.25 $$

16. $$ \lim _{x ightarrow 3} \frac{x^{2}-8 x+15}{x^{2}-7 x+12} = 2 $$

17. $$ \lim _{h ightarrow \frac{1}{2}} \frac{4 h^{2}+4 h-3}{2 h-1} = 4 $$

18. $$ \lim _{x ightarrow \frac{2}{3}} \frac{3 x-2}{3 x^{2}-11 x+6} = -\frac{3}{7} $$ o $$ -0.\dot{4}2857\dot{1} $$

19. $$ \lim _{w ightarrow -\frac{4}{3}} \frac{9 w^{2}+9 w-4}{3 w^{2}+7 w+4} = 15 $$

20. $$ \lim _{y ightarrow 6} \frac{2 y^{2}-15 y+18}{3 y^{2}-17 y-6} = \frac{9}{19} $$

21. $$ \lim _{x ightarrow 5} \frac{2 x^{2}-13 x+15}{x^{2}-x-20} = \frac{7}{9} $$ o $$ 0.\dot{7} $$

22. $$ \lim _{x ightarrow -\frac{1}{3}} \frac{9 x^{2}-1}{6 x^{2}+5 x+1} = -6 $$

23. $$ \lim _{y ightarrow -1} \frac{y+1}{y^{3}+1} = \frac{1}{3} $$ o $$ 0.\dot{3} $$

Si necesitas más detalles sobre algún cálculo específico, házmelo saber.
Aquí están los resultados de los límites: 15. $$ \frac{1}{4} $$ o $$ 0.25 $$ 16. $$ 2 $$ 17. $$ 4 $$ 18. $$ -\frac{3}{7} $$ o $$ -0.\dot{4}2857\dot{1} $$ 19. $$ 15 $$ 20. $$ \frac{9}{19} $$ 21. $$ \frac{7}{9} $$ o $$ 0.\dot{7} $$ 22. $$ -6 $$ 23. $$ \frac{1}{3} $$ o $$ 0.\dot{3} $$ Si necesitas más detalles sobre algún cálculo, avísame.

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