$\underset{t\to -2}{lim}\(t^2\+5t\)\=-6$$$

The limit of the first factor in the numerator exists.

 $\underset{t\to -2}{lim}\(t^{10}-6t^5\+2t\)\=1212$  $$

The limit of the second factor in the numerator exists.

 $\underset{t\to -2}{lim}\(t^4\+4t^2\+4\)\=36$  $$

The limit of the denominator exists and is not zero.

 $\underset{t\to -2}{lim}\(\(t^2\+5t\)\(t^{10}-6t^5\+2t\)\)$  = $-6\cdot 1212$  $$

The limit of the numerator is the product of the limits of the separate factors.

 $\underset{t\to -2}{lim}\frac{\(t^2\+5t\)\(t^{10}-6t^5\+2t\)}{t^4\+4t^2\+4}$  = $-\frac{6\cdot 1212}{36}$  = $-202$$$

The limit of the given rational function is the quotient of the limits because all appropriate limits exist, and the limit of the denominator is not zero.

Table 1.3.3(a)   Quotient rule for limits applied to $\underset{t\to -2}{lim}\frac{\left(t^2\+5t\right)\left(t^{10}-6t^5\+2t\right)}{t^4\+4t^2\+4}$

Annotated stepwise solution via the Context Panel

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$\underset{t\to -2}{lim}\frac{\left(t^2\+5t\right)\left(t^{10}-6t^5\+2t\right)}{t^4\+4t^2\+4}$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Limit Steps}\\ & & \underset{t\to \mathrm{-2}}{lim}\frac{\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{t^4+4t^2+4}\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{quotient}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{quotient}}\text{rule}\\ & & \underset{t\to a}{lim}\frac{f\left(t\right)}{g\left(t\right)}=\frac{\underset{t\to a}{lim}f\left(t\right)}{\underset{t\to a}{lim}g\left(t\right)}\\ & & f_1\left(t\right)=t^2+5t\\ & & f_2\left(t\right)=t^{10}-6t^5+2t\\ & & f_3\left(t\right)=\frac{1}{t^4+4t^2+4}\\ & & \text{This gives:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{\underset{t\to \mathrm{-2}}{lim}\left(t^4+4t^2+4\right)}\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \underset{t\to a}{lim}\left(f_1\left(t\right)+f_2\left(t\right)+f_3\left(t\right)\right)=\left(\underset{t\to a}{lim}{f}_1\left(t\right)\right)+\left(\underset{t\to a}{lim}{f}_2\left(t\right)\right)+\left(\underset{t\to a}{lim}{f}_3\left(t\right)\right)\\ & & f_1\left(t\right)=t^4\\ & & f_2\left(t\right)=4t^2\\ & & f_3\left(t\right)=4\\ & & \text{This gives:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{\left(\underset{t\to \mathrm{-2}}{lim}{t}^4\right)+\left(\underset{t\to \mathrm{-2}}{lim}4t^2\right)+\left(\underset{t\to \mathrm{-2}}{lim}4\right)}\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{constant}}\text{rule to the term}\underset{t\to \mathrm{-2}}{lim}4\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant}}\text{rule}\\ & & \mathrm{Limit}\left(C\,t\right)=C\\ & \text{◦}& \text{This means}\\ & & \underset{t\to \mathrm{-2}}{lim}4=4\\ & & \text{We can now rewrite the limit as:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{\left(\underset{t\to \mathrm{-2}}{lim}{t}^4\right)+\left(\underset{t\to \mathrm{-2}}{lim}4t^2\right)+4}\\ \text{▫}& & \text{4. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{t\to \mathrm{-2}}{lim}4t^2\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \underset{t\to \mathrm{-2}}{lim}Cf\left(t\right)=C\left(\underset{t\to \mathrm{-2}}{lim}f\left(t\right)\right)\\ & \text{◦}& \text{This means:}\\ & & \underset{t\to \mathrm{-2}}{lim}4t^2=4\cdot \left(\underset{t\to \mathrm{-2}}{lim}{t}^2\right)\\ & & \text{We can rewrite the limit as:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{\left(\underset{t\to \mathrm{-2}}{lim}{t}^4\right)+4\left(\underset{t\to \mathrm{-2}}{lim}{t}^2\right)+4}\\ \text{▫}& & \text{5. Apply the}\mathbf{\text{power}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \underset{t\to a}{lim}{t}^n={\left(\underset{t\to a}{lim}t\right)}^n\\ & & \text{This gives:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{\left(\underset{t\to \mathrm{-2}}{lim}{t}^4\right)+4{\left(\underset{t\to \mathrm{-2}}{lim}t\right)}^2+4}\\ \text{▫}& & \text{6. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{\left(\underset{t\to \mathrm{-2}}{lim}{t}^4\right)+20}\\ \text{▫}& & \text{7. Apply the}\mathbf{\text{power}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \underset{t\to a}{lim}{t}^n={\left(\underset{t\to a}{lim}t\right)}^n\\ & & \text{This gives:}\\ & & \frac{\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)}{{\left(\underset{t\to \mathrm{-2}}{lim}t\right)}^4+20}\\ \text{▫}& & \text{8. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & \frac{\left(\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\left(t^{10}-6t^5+2t\right)\right)}{36}\\ \text{▫}& & \text{9. Apply the}\mathbf{\text{product}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{product}}\text{rule}\\ & & \underset{t\to a}{lim}f\left(t\right)g\left(t\right)=\left(\underset{t\to a}{lim}f\left(t\right)\right)\left(\underset{t\to a}{lim}g\left(t\right)\right)\\ & & f\left(t\right)=t^2+5t\\ & & g\left(t\right)=t^{10}-6t^5+2t\\ & & \text{This gives:}\\ & & \frac{\left(\underset{t\to \mathrm{-2}}{lim}\left(t^2+5t\right)\right)\left(\underset{t\to \mathrm{-2}}{lim}\left(t^{10}-6t^5+2t\right)\right)}{36}\\ \text{▫}& & \text{10. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \underset{t\to a}{lim}\left(f\left(t\right)+g\left(t\right)\right)=\left(\underset{t\to a}{lim}f\left(t\right)\right)+\left(\underset{t\to a}{lim}g\left(t\right)\right)\\ & & f\left(t\right)=t^2\\ & & g\left(t\right)=5t\\ & & \text{This gives:}\\ & & \frac{\left(\left(\underset{t\to \mathrm{-2}}{lim}{t}^2\right)+\left(\underset{t\to \mathrm{-2}}{lim}5t\right)\right)\left(\underset{t\to \mathrm{-2}}{lim}\left(t^{10}-6t^5+2t\right)\right)}{36}\\ \text{▫}& & \text{11. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{t\to \mathrm{-2}}{lim}5t\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \underset{t\to \mathrm{-2}}{lim}Cf\left(t\right)=C\left(\underset{t\to \mathrm{-2}}{lim}f\left(t\right)\right)\\ & \text{◦}& \text{This means:}\\ & & \underset{t\to \mathrm{-2}}{lim}5t=5\cdot \left(\underset{t\to \mathrm{-2}}{lim}t\right)\\ & & \text{We can rewrite the limit as:}\\ & & \frac{\left(\left(\underset{t\to \mathrm{-2}}{lim}{t}^2\right)+5\left(\underset{t\to \mathrm{-2}}{lim}t\right)\right)\left(\underset{t\to \mathrm{-2}}{lim}\left(t^{10}-6t^5+2t\right)\right)}{36}\\ \text{▫}& & \text{12. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & \frac{\left(\left(\underset{t\to \mathrm{-2}}{lim}{t}^2\right)-10\right)\left(\underset{t\to \mathrm{-2}}{lim}\left(t^{10}-6t^5+2t\right)\right)}{36}\\ \text{▫}& & \text{13. Apply the}\mathbf{\text{power}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \underset{t\to a}{lim}{t}^n={\left(\underset{t\to a}{lim}t\right)}^n\\ & & \text{This gives:}\\ & & \frac{\left({\left(\underset{t\to \mathrm{-2}}{lim}t\right)}^2-10\right)\left(\underset{t\to \mathrm{-2}}{lim}\left(t^{10}-6t^5+2t\right)\right)}{36}\\ \text{▫}& & \text{14. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}\left(t^{10}-6t^5+2t\right)\right)}{6}\\ \text{▫}& & \text{15. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \underset{t\to a}{lim}\left(f_1\left(t\right)+f_2\left(t\right)+f_3\left(t\right)\right)=\left(\underset{t\to a}{lim}{f}_1\left(t\right)\right)+\left(\underset{t\to a}{lim}{f}_2\left(t\right)\right)+\left(\underset{t\to a}{lim}{f}_3\left(t\right)\right)\\ & & f_1\left(t\right)=t^{10}\\ & & f_2\left(t\right)=-6t^5\\ & & f_3\left(t\right)=2t\\ & & \text{This gives:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}{t}^{10}\right)}{6}-\frac{\left(\underset{t\to \mathrm{-2}}{lim}\left(-6t^5\right)\right)}{6}-\frac{\left(\underset{t\to \mathrm{-2}}{lim}2t\right)}{6}\\ \text{▫}& & \text{16. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{t\to \mathrm{-2}}{lim}\left(-6t^5\right)\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \underset{t\to \mathrm{-2}}{lim}Cf\left(t\right)=C\left(\underset{t\to \mathrm{-2}}{lim}f\left(t\right)\right)\\ & \text{◦}& \text{This means:}\\ & & \underset{t\to \mathrm{-2}}{lim}\left(-6t^5\right)=\left(\mathrm{-6}\right)\cdot \left(\underset{t\to \mathrm{-2}}{lim}{t}^5\right)\\ & & \text{We can rewrite the limit as:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}{t}^{10}\right)}{6}+\left(\underset{t\to \mathrm{-2}}{lim}{t}^5\right)-\frac{\left(\underset{t\to \mathrm{-2}}{lim}2t\right)}{6}\\ \text{▫}& & \text{17. Apply the}\mathbf{\text{power}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \underset{t\to a}{lim}{t}^n={\left(\underset{t\to a}{lim}t\right)}^n\\ & & \text{This gives:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}{t}^{10}\right)}{6}+{\left(\underset{t\to \mathrm{-2}}{lim}t\right)}^5-\frac{\left(\underset{t\to \mathrm{-2}}{lim}2t\right)}{6}\\ \text{▫}& & \text{18. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}{t}^{10}\right)}{6}-32-\frac{\left(\underset{t\to \mathrm{-2}}{lim}2t\right)}{6}\\ \text{▫}& & \text{19. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{t\to \mathrm{-2}}{lim}2t\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \underset{t\to \mathrm{-2}}{lim}Cf\left(t\right)=C\left(\underset{t\to \mathrm{-2}}{lim}f\left(t\right)\right)\\ & \text{◦}& \text{This means:}\\ & & \underset{t\to \mathrm{-2}}{lim}2t=2\cdot \left(\underset{t\to \mathrm{-2}}{lim}t\right)\\ & & \text{We can rewrite the limit as:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}{t}^{10}\right)}{6}-32-\frac{\left(\underset{t\to \mathrm{-2}}{lim}t\right)}{3}\\ \text{▫}& & \text{20. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & -\frac{\left(\underset{t\to \mathrm{-2}}{lim}{t}^{10}\right)}{6}-\frac{94}{3}\\ \text{▫}& & \text{21. Apply the}\mathbf{\text{power}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{power}}\text{rule}\\ & & \underset{t\to a}{lim}{t}^n={\left(\underset{t\to a}{lim}t\right)}^n\\ & & \text{This gives:}\\ & & -\frac{{\left(\underset{t\to \mathrm{-2}}{lim}t\right)}^{10}}{6}-\frac{94}{3}\\ \text{▫}& & \text{22. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{t\to a}{lim}t=a\\ & & \text{This gives:}\\ & & \mathrm{-202}\end{array}$$$

Table 1.3.3(b)   Maple's stepwise solution via the All Solution Steps option in the Context Panel