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Worksheet: Finding the Limits of Functions of Two or More Variables

In this worksheet, we will practice finding the limit of a function in more than one variable as we approach a point in the domain of the function.

Q1:

Evaluate the limit, if it exists.

Q2:

Evaluate the limit, if it exists.

  • A1
  • B0
  • C βˆ’ 1
  • DThe limit does not exist

Q3:

Evaluate l i m ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 3 3 2 2 π‘₯ βˆ’ 𝑦 π‘₯ + π‘₯ 𝑦 + 𝑦 , if it exists.

Q4:

Evaluate the limit l i m ( π‘₯ , 𝑦 ) β†’ ( 1 , 0 ) 2 2 π‘₯ 𝑦 βˆ’ 𝑦 ( π‘₯ βˆ’ 1 ) + 𝑦 , if it exists.

  • A 1 2
  • B0
  • CThe limit does not exist.

Q5:

Evaluate the limit l i m ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 4 2 8 π‘₯ 𝑦 π‘₯ + 𝑦 , if it exists.

  • A1
  • B0
  • C 1 2
  • DThe limit does not exist.

Q6:

Evaluate the limit, if it exists.

Q7:

Evaluate the limit, if it exists.

  • A βˆ’ 2
  • B0
  • C βˆ’ 4 3
  • DThe limit does not exist

Q8:

Evaluate the limit, if it exists.

Q9:

Evaluate the limit l i m ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 2 2 2 2 π‘₯ + 𝑦 √ π‘₯ + 𝑦 + 1 βˆ’ 1 , if it exists.

Q10:

Evaluate the limit, if it exists.

  • A0
  • B βˆ’ πœ‹
  • CThe limit does not exist
  • D πœ‹ 2

Q11:

Evaluate l i m c o s ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) ο€½ 1 π‘₯ 𝑦  , if it exists.

  • A1
  • B0
  • C βˆ’ 1
  • DThe limit does not exist.

Q12:

Evaluate the limit l i m c o s ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 2 2 ο€Ή π‘₯ + 𝑦  ο€½ 1 π‘₯ 𝑦  , if it exists.

  • A 1 2
  • B1
  • CThe limit does not exist.

Q13:

Evaluate the limit, if it exists.

  • A
  • B0
  • C1
  • DThe limit does not exist
  • E

Q14:

Evaluate the limit l i m s i n ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 2 2 4 4 𝑦 π‘₯ π‘₯ + 𝑦 , if it exists.

  • A 1 2
  • B0
  • CThe limit does not exist.

Q15:

Evaluate l i m c o s ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 4 2 4 4 5 𝑦 π‘₯ π‘₯ + 𝑦 if it exists.

  • A5
  • B0
  • C 5 2
  • DThe limit does not exist.

Q16:

Evaluate the limit, if it exists.

Q17:

Evaluate the limit, if it exists.

  • A 𝑒 √ 6
  • B 𝑒
  • C 𝑒 √ 1 5
  • D 𝑒 2
  • EThe limit does not exist

Q18:

Evaluate the limit, if it exists.

  • A0
  • BThe limit does not exist
  • C √ 3

Q19:

Evaluate the limit, if it exists.

Q20:

Evaluate l i m ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 2 2 2 π‘₯ 𝑦 π‘₯ + 𝑦 , if it exists.

Q21:

Evaluate l i m ( π‘₯ , 𝑦 ) β†’ ( 1 , βˆ’ 1 ) 2 2 π‘₯ βˆ’ 2 π‘₯ 𝑦 + 𝑦 π‘₯ βˆ’ 𝑦 , if it exists.

Q22:

Evaluate the limit l i m ( π‘₯ , 𝑦 ) β†’ ( 0 , 0 ) 2 2 4 π‘₯ 𝑦 π‘₯ + 𝑦 , if it exists.

  • A 1 2
  • B0
  • C βˆ’ 1 2
  • DThe limit does not exist.

Q23:

Evaluate the limit, if it exists.

Q24:

Evaluate the limit, if it exists.

  • A βˆ’ 1
  • B1
  • C0
  • DThe limit does not exist

Q25:

Evaluate the limit, if it exists.

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