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Section 13.1 : Limits

Evaluate each of the following limits.

  1. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {2,1,0} \right)} \frac{{\left( {4y - {z^3}} \right){{\bf{e}}^{3x - 6}}}}{{4z - y{x^2}}}\)
  2. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {3, - 7} \right)} \frac{{6x - y + xy}}{{2{x^3} + {y^3}}}\)
  3. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( { - 3,4} \right)} \frac{{4{x^2} - xy - 3{y^2}}}{{12{x^2} + 17xy + 6{y^2}}}\)
  4. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( { - 1,10} \right)} \frac{{10{x^2} + 11xy + {y^2}}}{{10{x^2} - 39xy - 4{y^2}}}\)
  5. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{2{x^2} + 7{y^2}}}{{4{y^2} + {x^2}}}\)
  6. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{6\,\,\sqrt[3]{x} - 3{y^{10}}}}{{9{y^{30}} + 2x}}\)
  7. \(\displaystyle \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{2{x^4}y}}{{{x^8} + 6{y^2}}}\)