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Worksheet: 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 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.

Q3:

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.

Q4:

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

Q5:

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.

Q6:

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

Q7:

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.

Q8:

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.

Q9:

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

Q10:

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.

Q11:

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.

Q12:

Evaluate the limit l i m s i n (  οŽ•  ) β†’ (  οŽ•  ) οŠͺ   𝑦 π‘₯ 𝑦 π‘₯ + 𝑦 , if it exists.

Q13:

Evaluate the limit l i m (  οŽ•  ) β†’ (  οŽ•  )     π‘₯ βˆ’ 𝑦 π‘₯ + 𝑦 , if it exists.

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

Q14:

Evaluate l i m (  οŽ•  ) β†’ (  οŽ•  )    π‘₯ 𝑦 π‘₯ + 𝑦 , if it exists.

Q15:

Evaluate l i m (  οŽ•  ) β†’ (  οŽ•  ) οŠͺ    π‘₯ βˆ’ 4 𝑦 π‘₯ + 2 𝑦 , if it exists.

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

Q16:

Evaluate l i m ( π‘₯ , 𝑦 ) β†’ ( 3 , 2 ) 2 3 2 ο€Ή π‘₯ 𝑦 βˆ’ 4 𝑦  , if it exists.

Q17:

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

Q18:

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

Q19:

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.

Q20:

Evaluate l i m t a n ( π‘₯ , 𝑦 , 𝑧 ) β†’ ( πœ‹ , 0 , ) 𝑦 1 3 2 𝑒 π‘₯ 𝑧 , if it exists.

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

Q21:

Evaluate the limit, if it exists.

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

Q22:

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

Q23:

Evaluate l i m s i n ( π‘₯ , 𝑦 ) β†’ ( πœ‹ , ) πœ‹ 2 𝑦 ( π‘₯ βˆ’ 𝑦 ) , if it exists.

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

Q24:

Evaluate the limit, if it exists.

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

Q25:

Evaluate the limit l i m (  οŽ•  ) β†’ (  οŽ•   ) √     𝑒 , if it exists.

  • A 𝑒 √ 
  • B 𝑒
  • C 𝑒 √  
  • D 𝑒 
  • EThe limit does not exist.

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