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 𝑂𝑃 is inclined at 𝜃 radians.

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

  • A(1,𝜃)cot
  • B(1,𝜃)cos
  • C(1,𝜃)sin
  • D(1,𝜃)tan

Write the following inequalities in terms of sin𝜃, 𝜃, and cos𝜃: 𝑄𝑅<𝑇𝑄<𝑇𝑃.lengthofarcfromto

  • Acossincos𝜃<𝜃<𝜃𝜃
  • Bsincot𝜃<𝜃<𝜃
  • Csinsincos𝜃<𝜃<𝜃𝜃
  • Dsincoscos𝜃𝜃<𝜃<𝜃

By dividing your inequalities by sin𝜃, using the squeeze theorem and the fact that limcos𝜃=1, which of the following conclusions can you draw?

  • Alimsin𝜃𝜃 does not exist.
  • Blimsin𝜃𝜃=0
  • Climsin𝜃𝜃=1

Q2:

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

  • Alim𝑔(𝑥)=1
  • Blimlimlim(𝑥)<𝑔(𝑥)<𝑓(𝑥)
  • Clim𝑔(𝑥)=3
  • Dlim𝑔(𝑥)=2
  • EThe limit of 𝑔(𝑥) as 𝑥3 does not exist.

Q3:

The graph shown is that of the function 𝑓(𝑥)=𝑥2𝜋𝑥sin.

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 lim𝑓(𝑥)=0?

  • A𝐿(𝑥)=|𝑥|0.001
  • B𝐿(𝑥)=2|𝑥|
  • C𝐿(𝑥)=𝑥
  • D𝐿(𝑥)=2
  • E𝐿(𝑥)=𝑥sin

Q4:

In deciding that limsin𝑥1𝑥=0, which assumptions do we use? Select all valid answers.

  • A11𝑥1𝑥=0sinandlim
  • Blimsin1𝑥=1 and lim𝑥=0
  • Climsin1𝑥 does not exist.
  • Dlimsin1𝑥=1 and lim𝑥=0
  • Elimsin1𝑥=0

Q5:

Calculate limcos𝑥2𝑥 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)?

  • Alim𝑓(𝑥)=2
  • Blim𝑓(𝑥)=1
  • Clim𝑓(𝑥)=12
  • Dlim𝑓(𝑥)=3
  • EThe limit does not exist.

Q7:

Knowing that the squeeze theorem applies when the limit is taken at , determine limsin(3𝑥)𝑥.

Q8:

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

If 3𝑥3𝑔(𝑥)2𝑥4𝑥+3, then lim𝑔(𝑥)=0.

  • ATrue
  • BFalse

Q9:

Use the squeeze theorem to evaluate limsin3𝑥𝜋𝑥.

Q10:

Use the squeeze theorem to evaluate limcos2𝜃1𝜃.

Q11:

Using the squeeze theorem, calculate limcos3𝑥30𝑥+751𝑥5.

Q12:

Given that 3𝑥+9𝑥5𝑓(𝑥)𝑥3𝑥+4, find lim𝑓(𝑥).

  • AThe limit does not exist.
  • B1
  • C4
  • D5
  • E74

Q13:

Calculate limsin(2𝑥)5𝑥+3 using the squeeze theorem.

Q14:

Evaluate the limit limcos5+(4|(𝑛!)|).

Q15:

Given the function 𝑓(𝑥), such that 2𝑥5𝑥+7𝑓(𝑥)4𝑥10𝑥2 for all 𝑥, find lim5𝑥𝑓(𝑥)3.

  • A23
  • B35
  • CThe limit does not exist.
  • D0
  • E25

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