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Section 4.10 : L'Hospital's Rule and Indeterminate Forms

Use L’Hospital’s Rule to evaluate each of the following limits.

  1. \(\displaystyle \mathop {\lim }\limits_{x \to 2} \frac{{{x^3} - 7{x^2} + 10x}}{{{x^2} + x - 6}}\) Solution
  2. \(\displaystyle \mathop {\lim }\limits_{w \to \, - 4} \frac{{\sin \left( {\pi w} \right)}}{{{w^2} - 16}}\) Solution
  3. \(\displaystyle \mathop {\lim }\limits_{t \to \infty } \frac{{\ln \left( {3t} \right)}}{{{t^2}}}\) Solution
  4. \(\displaystyle \mathop {\lim }\limits_{z \to 0} \frac{{\sin \left( {2z} \right) + 7{z^2} - 2z}}{{{z^2}{{\left( {z + 1} \right)}^2}}}\) Solution
  5. \(\displaystyle \mathop {\lim }\limits_{x \to \, - \infty } \frac{{{x^2}}}{{{{\bf{e}}^{1 - \,x}}}}\) Solution
  6. \(\displaystyle \mathop {\lim }\limits_{z \to \infty } \frac{{{z^2} + {{\bf{e}}^{4\,z}}}}{{2z - {{\bf{e}}^{\,z}}}}\) Solution
  7. \(\mathop {\lim }\limits_{t \to \infty } \left[ {t\ln \left( {1 + \displaystyle \frac{3}{t}} \right)} \right]\) Solution
  8. \(\mathop {\lim }\limits_{w \to {0^ + }} \left[ {{w^2}\ln \left( {4{w^2}} \right)} \right]\) Solution
  9. \(\mathop {\lim }\limits_{x \to {1^ + }} \left[ {\left( {x - 1} \right)\tan \left( {\displaystyle \frac{\pi }{2}x} \right)} \right]\) Solution
  10. \(\mathop {\lim }\limits_{y \to {0^ + }} {\left[ {\cos \left( {2y} \right)} \right]^{{}^{1}/{}_{{{y^{\,2}}}}}}\) Solution
  11. \(\mathop {\lim }\limits_{x \to \infty } {\left[ {{{\bf{e}}^x} + x} \right]^{{}^{1}/{}_{x}}}\) Solution