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### Section 4-10 : L'Hospital's Rule and Indeterminate Forms

For problems 1 – 18 use L’Hospital’s Rule to evaluate the given limit.

1. $$\displaystyle \mathop {\lim }\limits_{x \to - 4} \frac{{{x^3} + 6{x^2} - 32}}{{{x^3} + 5{x^2} + 4x}}$$
2. $$\displaystyle \mathop {\lim }\limits_{w \to \, - \infty } \frac{{{{\bf{e}}^{ - 6w}}}}{{4 + {{\bf{e}}^{ - 3w}}}}$$
3. $$\displaystyle \mathop {\lim }\limits_{t \to 0} \frac{{\sin \left( {6t} \right)}}{{\sin \left( {11t} \right)}}$$
4. $$\displaystyle \mathop {\lim }\limits_{x \to 1} \frac{{{x^2} + 8x - 9}}{{{x^3} - 2{x^2} - 5x + 6}}$$
5. $$\displaystyle \mathop {\lim }\limits_{t \to 2} \frac{{{t^3} - 7{t^2} + 16t - 12}}{{{t^4} - 4{t^3} + 4{t^2}}}$$
6. $$\displaystyle \mathop {\lim }\limits_{w \to - \infty } \frac{{{w^2} - 4w + 1}}{{3{w^2} + 7w - 4}}$$
7. $$\displaystyle \mathop {\lim }\limits_{y \to \infty } \frac{{{y^2} - {{\bf{e}}^{6\,y}}}}{{4{y^2} + {{\bf{e}}^{7\,y}}}}$$
8. $$\displaystyle \mathop {\lim }\limits_{x \to 0} \frac{{2\cos \left( {4x} \right) - 4{x^2} - 2}}{{\sin \left( {2x} \right) - {x^2} - 2x}}$$
9. $$\displaystyle \mathop {\lim }\limits_{x \to \, - 3} \frac{{3{{\bf{e}}^{2\,x + 6}} + {x^2} - 12}}{{{x^3} + 6{x^2} + 9x}}$$
10. $$\displaystyle \mathop {\lim }\limits_{z \to 6} \frac{{\sin \left( {\pi z} \right)}}{{\ln \left( {z - 5} \right)}}$$
11. $$\mathop {\lim }\limits_{w \to \infty } \left[ {w\ln \left( {1 - \displaystyle \frac{2}{{3w}}} \right)} \right]$$
12. $$\mathop {\lim }\limits_{t \to {0^ + }} \left[ {\ln \left( t \right)\sin \left( t \right)} \right]$$
13. $$\mathop {\lim }\limits_{z \to - \infty } {z^2}{{\bf{e}}^z}$$
14. $$\mathop {\lim }\limits_{x \to \infty } \left[ {x\sin \left( \frac{7}{x}} \right)} \right$$
15. $$\mathop {\lim }\limits_{z \to {0^ + }} \left[ {{z^2}{{\left( {\ln z} \right)}^2}} \right]$$
16. $$\mathop {\lim }\limits_{x \to {0^ + }} {x^{{}^{1}/{}_{x}}}$$
17. $$\mathop {\lim }\limits_{t \to {0^ + }} {\left[ {{{\bf{e}}^t} + t} \right]^{{}^{1}/{}_{t}}}$$
18. $$\mathop {\lim }\limits_{x \to \infty } {\left[ {{{\bf{e}}^{ - 2x}} + 3x} \right]^{{}^{1}/{}_{x}}}$$
19. Suppose that we know that $$f'\left( x \right)$$ is a continuous function. Use L’Hospital’s Rule to show that, $\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( {x - h} \right)}}{{2h}} = f'\left( x \right)$
20. Suppose that we know that $$f''\left( x \right)$$ is a continuous function. Use L’Hospital’s Rule to show that, $\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - 2f\left( x \right) + f\left( {x - h} \right)}}{{{h^2}}} = f''\left( x \right)$