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Section 2: Limits and Continuity

Instructions

  • First, you should watch the concepts videos below explaining the topics in the section. 
  • Second, you should attempt to solve the exercises and then watch the videos explaining the exercises. 
  • Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
  • When you have finished the material below, you can start on Section 3 or return to the main several variable calculus page.

Concepts

  • Calculating the limit of a surface
  • The definition of the limit of a two-variable function
  • Limits at infinity and infinite limits of two-variable functions
 

Exercises


Directions: You should attempt to solve the problems first and then watch the video to see the solution. 
  1. Find the limit. If the limit does not exist, support your answer.
    1. \(\displaystyle \lim_{(x,y)\rightarrow (1,2)} \ln \left(1+x^2y^2\right)\)
    2. \(\displaystyle \lim_{(​x,y)\rightarrow (0,0)} \frac{x^3+xy^2}{x^2+y^2}\)

      1. \(\displaystyle \lim_{(x,y)\rightarrow (1,2)} \ln \left(1+x^2y^2\right)=\ln 5\)
      2. \(\displaystyle \lim_{(​x,y)\rightarrow (0,0)} \frac{x^3+xy^2}{x^2+y^2}=0\)


  2. Find the limit. If the limit does not exist, support your answer.\[ \lim_{(x,y)\rightarrow (0,0)} \frac{\sqrt{x^2+y^2+1}-1}{x^2+y^2}\]

    \(\displaystyle \lim_{(x,y)\rightarrow (0,0)} \frac{\sqrt{x^2+y^2+1}-1}{x^2+y^2}=\frac{1}{2}\)


  3. Find the limit. If the limit does not exist, support your answer. \[\lim_{(x,y)\rightarrow (0,0)}\frac{3xy}{x^2+y^2}\]

    The limit does not exist since the limit along \(y=0\) does not equal the limit along the path \(y=x\).


  4. Find the limit. If the limit does not exist, support your answer. \[\lim_{(x,y)\rightarrow (0,0)} \frac{x^2-y^2}{x^2+y^2}\]

    The limit does not exist since the limit along the \(x\)-axis does not equal the limit along the \(y\)-axis.


 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. When we want to study the \(z\) values of a surface \(z=f(x,y)\) near a point \((a,b)\) in the \(xy\)-plane using a table of values,  how can we determine whether \(\lim\limits_{(x,y)\to (a,b)} f(x,y)\) exists? If the limit does exist, how can we estimate the value of the limit?
  2. Does \(f(a,b)\) have to exist in order for \(\lim\limits_{(x,y)\to (a,b)} f(x,y)\) to exist?
  3. How can we prove \(\lim\limits_{(x,y)\to (a,b)} f(x,y)\) does not exist?