Worksheet: Limits of Multivariable Functions

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In this worksheet, we will practice finding the limit of multivariable functions as we approach a point in the domain of the function.

Q1:

Evaluate the limit l i m s i n ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 2 2 4 4 𝑦 𝑥 𝑥 + 𝑦 , if it exists.

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

Q2:

Evaluate l i m ( 𝑥 , 𝑦 ) → ( 3 , 2 ) 2 3 2 𝑥 𝑦 − 4 𝑦 , if it exists.

Q3:

Evaluate the limit, if it exists.

  • A1
  • B0
  • C − 1
  • DThe limit does not exist

Q4:

Evaluate l i m ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 3 3 2 2 𝑥 − 𝑦 𝑥 + 𝑥 𝑦 + 𝑦 , if it exists.

Q5:

Evaluate the limit l i m ( 𝑥 , 𝑦 ) → ( 1 , 0 ) 2 2 𝑥 𝑦 − 𝑦 ( 𝑥 − 1 ) + 𝑦 , if it exists.

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

Q6:

Evaluate the limit l i m ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 4 2 8 𝑥 𝑦 𝑥 + 𝑦 , if it exists.

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

Q7:

Evaluate l i m ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 4 4 4 𝑥 𝑦 𝑥 + 𝑦 , if it exists.

Q8:

Evaluate l i m ( ) → ( ) 𝑥 − 4 𝑦 𝑥 + 2 𝑦 , if it exists.

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

Q9:

Evaluate l i m ( 𝑥 , 𝑦 , 𝑧 ) → ( 0 , 0 , 0 ) 2 2 2 2 2 2 𝑥 𝑦 𝑧 𝑥 + 𝑦 + 𝑧 , if it exists.

Q10:

Evaluate the limit, if it exists.

  • A0
  • BThe limit does not exist
  • C1
  • D2

Q11:

Evaluate l i m s i n ( 𝑥 , 𝑦 ) → ( 𝜋 , ) 𝜋 2 𝑦 ( 𝑥 − 𝑦 ) , if it exists.

  • A0
  • B − 𝜋
  • CThe limit does not exist.
  • D 𝜋 2

Q12:

Evaluate l i m c o s ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 1 𝑥 𝑦 , if it exists.

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

Q13:

Evaluate the limit l i m c o s ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 2 2 𝑥 + 𝑦 1 𝑥 𝑦 , if it exists.

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

Q14:

Evaluate l i m c o s ( ) → ( ) 𝑥 𝑦 𝑦 𝑥 + 𝑦 , if it exists.

  • A 1 2
  • B0
  • C1
  • DThe limit does not exist.
  • E 4 𝜋

Q15:

Evaluate l i m c o s ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 4 2 4 4 5 𝑦 𝑥 𝑥 + 𝑦 if it exists.

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

Q16:

Evaluate l i m ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 𝑥 𝑦 𝑒 , if it exists.

Q17:

Evaluate the limit l i m ( ) → ( ) √ 𝑒 , if it exists.

  • A 𝑒 √
  • B 𝑒
  • C 𝑒 √
  • D 𝑒
  • EThe limit does not exist.

Q18:

Evaluate l i m t a n ( 𝑥 , 𝑦 , 𝑧 ) → ( 𝜋 , 0 , ) 𝑦 1 3 2 𝑒 𝑥 𝑧 , if it exists.

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

Q19:

Evaluate if it exists.

Q20:

Evaluate l i m ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 2 2 2 𝑥 𝑦 𝑥 + 𝑦 , if it exists.

Q21:

Evaluate l i m ( 𝑥 , 𝑦 ) → ( 1 , − 1 ) 2 2 𝑥 − 2 𝑥 𝑦 + 𝑦 𝑥 − 𝑦 , if it exists.

Q22:

Evaluate the limit l i m ( 𝑥 , 𝑦 ) → ( 0 , 0 ) 2 2 4 𝑥 𝑦 𝑥 + 𝑦 , if it exists.

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

Q23:

Evaluate the limit l i m s i n ( ) → ( ) 𝑦 𝑥 𝑦 𝑥 + 𝑦 , if it exists.

Q24:

Evaluate the limit l i m ( ) → ( ) 𝑥 − 𝑦 𝑥 + 𝑦 , if it exists.

  • A − 1
  • B1
  • C0
  • DThe limit does not exist.

Q25:

Evaluate l i m ( ) → ( ) 𝑥 𝑦 𝑥 + 𝑦 , if it exists.

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