Worksheet: The Squeeze Theorem

In this worksheet, we will practice using the squeeze (sandwich) theorem to evaluate some limits when the value of a function is bounded by the values of two other functions.

Q1:

Consider the following arc of a unit circle, where ray 𝑂 𝑃 is inclined at 𝜃 radians.

What, in terms of 𝜃 , are the coordinates of 𝑃 ?

  • A ( 1 , 𝜃 ) s i n
  • B ( 1 , 𝜃 ) c o s
  • C ( 1 , 𝜃 ) c o t
  • D ( 1 , 𝜃 ) t a n

Write the following inequalities in terms of s i n 𝜃 , 𝜃 , and c o s 𝜃 : 𝑄 𝑅 < 𝑇 𝑄 < 𝑇 𝑃 . l e n g t h o f a r c f r o m t o

  • A s i n s i n c o s 𝜃 < 𝜃 < 𝜃 𝜃
  • B s i n c o t 𝜃 < 𝜃 < 𝜃
  • C s i n c o s c o s 𝜃 𝜃 < 𝜃 < 𝜃
  • D c o s s i n c o s 𝜃 < 𝜃 < 𝜃 𝜃

By dividing your inequalities by s i n 𝜃 , using the squeeze theorem and the fact that l i m c o s 𝜃 = 1 , which of the following conclusions can you draw?

  • A l i m s i n 𝜃 𝜃 = 0
  • B l i m s i n 𝜃 𝜃 does not exist.
  • C l i m s i n 𝜃 𝜃 = 1

Q2:

Given that the function 𝑔 is continuous, which of the following can we conclude from the squeeze theorem?

  • A l i m 𝑔 ( 𝑥 ) = 3
  • BThe limit of 𝑔 ( 𝑥 ) as 𝑥 3 does not exist.
  • C l i m 𝑔 ( 𝑥 ) = 2
  • D l i m 𝑔 ( 𝑥 ) = 1
  • E l i m l i m l i m ( 𝑥 ) < 𝑔 ( 𝑥 ) < 𝑓 ( 𝑥 )

Q3:

The graph shown is that of the function 𝑓 ( 𝑥 ) = 𝑥 2 𝜋 𝑥 s i n .

Given that 𝑈 ( 𝑥 ) = | 𝑥 | is a function such that 𝑈 ( 𝑥 ) 𝑓 ( 𝑥 ) for all 𝑥 , which of the following functions can be 𝐿 ( 𝑥 ) such that 𝑓 ( 𝑥 ) 𝐿 ( 𝑥 ) , so that we can show that l i m 𝑓 ( 𝑥 ) = 0 ?

  • A 𝐿 ( 𝑥 ) = 2 | 𝑥 |
  • B 𝐿 ( 𝑥 ) = | 𝑥 | 0 . 0 0 1
  • C 𝐿 ( 𝑥 ) = 𝑥 s i n
  • D 𝐿 ( 𝑥 ) = 2
  • E 𝐿 ( 𝑥 ) = 𝑥

Q4:

In deciding that l i m s i n 𝑥 1 𝑥 = 0 , which assumptions do we use? Select all valid answers.

  • A 1 1 𝑥 1 𝑥 = 0 s i n a n d l i m
  • B l i m s i n 1 𝑥 does not exist.
  • C l i m s i n 1 𝑥 = 1 and l i m 𝑥 = 0
  • D l i m s i n 1 𝑥 = 1 and l i m 𝑥 = 0
  • E l i m s i n 1 𝑥 = 0

Q5:

Calculate l i m c o s 𝑥 2 𝑥 using the squeeze theorem.

Q6:

The figure shows the graphs of functions 𝐴 and 𝐵 with 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) for 𝑥 between 2 and 3.8.

What does the squeeze theorem tell us about a continuous function 𝑓 whose graph lies in the shaded region over the interval ( 2 , 3 . 8 ) ?

  • A l i m 𝑓 ( 𝑥 ) = 2
  • B l i m 𝑓 ( 𝑥 ) = 1 2
  • CThe limit does not exist.
  • D l i m 𝑓 ( 𝑥 ) = 1
  • E l i m 𝑓 ( 𝑥 ) = 3

Q7:

Knowing that the squeeze theorem applies when the limit is taken at , determine l i m s i n ( 3 𝑥 ) 𝑥 .

Q8:

Using the squeeze theorem, check whether the following statement is true or false:

If 3 𝑥 3 𝑔 ( 𝑥 ) 2 𝑥 4 𝑥 + 3 , then l i m 𝑔 ( 𝑥 ) = 0 .

  • AFalse
  • BTrue

Q9:

Use the squeeze theorem to evaluate l i m s i n 3 𝑥 𝜋 𝑥 .

Q10:

Use the squeeze theorem to evaluate l i m c o s 2 𝜃 1 𝜃 .

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