Worksheet: Formal Definition of a Limit

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In this worksheet, we will practice interpreting the formal epsilon–delta definition of a limit and finding the value of delta for a given epsilon.

Q1:

Find the largest 𝛿>0 such that if |𝑥−5|<𝛿, then |||1𝑥−15|||<110. Give your answer as a fraction.

  • A 5 3
  • B 1 2 5 5
  • C 5 3 3
  • D 2 5 3
  • E 4 7 1 0

Q2:

Find the largest 𝛿>0 such that if |𝑥−5|<𝛿, then |||1𝑥−15|||<𝜀. Give your answer as a fraction involving 𝜀.

  • A 5 5 𝜀 + 1
  • B 2 5 𝜀 + 1 0 5 𝜀 + 1
  • C 2 5 𝜀 5 𝜀 + 1
  • D − 2 5 𝜀 + 1 0 5 𝜀 + 1
  • E − 2 5 𝜀 5 𝜀 + 1

Q3:

Find the largest 𝑎>0 such that if |𝑥−𝑎|<𝑎, then |||1𝑥−1𝑎|||<𝜀. Give your answer as a fraction involving 𝜀 and 𝑎.

  • A 𝑎 𝜀 + 2 𝑎 𝑎 𝜀 + 1
  • B 𝑎 𝜀 − 1 𝑎 𝜀
  • C 𝑎 𝜀 + 1 𝑎 𝜀
  • D 𝑎 𝜀 𝑎 𝜀 + 1
  • E 𝑎 𝜀 − 2 𝑎 𝑎 𝜀 + 1

Q4:

In the figure, we have the graph 𝑦=𝑓(𝑥) that is increasing and concave up. Next to (𝑥,𝑦) are the points (𝑎,𝑦+𝑘) and (𝑏,𝑦−𝑘) for a small 𝑘>0.

Which point is closer to 𝑥 along the horizontal axis, 𝑎 or 𝑏?

  • A 𝑏
  • B 𝑎

What is the distance 𝛿 between the nearest point and 𝑥 from your answer above? Give an expression that involves 𝑓 and absolute values.

  • A | | 𝑥 − 𝑓 ( 𝑓 ( 𝑥 ) + 𝑘 ) | |
  • B | | 𝑥 + 𝑓 ( 𝑓 ( 𝑥 ) − 𝑘 ) | |
  • C | | 𝑥 − 𝑓 ( 𝑓 ( 𝑥 ) − 𝑘 ) | |
  • D | | 𝑥 + 𝑓 ( 𝑓 ( 𝑥 ) + 𝑘 ) | |

Use your answers above to find 𝛿 such that |𝑒−1|<0.1 whenever |𝑥−0|<𝛿. Give your answer to 4 decimal places.

Q5:

The figure shows the graph of 𝑓(𝑥)=2𝑥 around a point (𝑥,𝑦) where 𝑥>0 and 𝑦=𝑓(𝑥). Nearby are points 𝑎,𝑦+12 and 𝑏,𝑦−12.

From the graph, looking at the 𝑥-axis, which of the points 𝑎 and 𝑏 is nearest to 𝑥? Let this number be 𝑝.

  • A 𝑏
  • B 𝑎

What is 𝑎 in terms of 𝑥?

  • A 4 + 𝑥 2 𝑥
  • B 2 𝑥 4 + 𝑥
  • C 4 + 𝑥 4 𝑥
  • D 4 𝑥
  • E 4 𝑥 4 + 𝑥

The claim is that if 𝑥>0 and 2𝑥>12, then |||2𝑥−2𝑥|||<12∗ provided that |𝑥−𝑥|<𝛿∗ is true for all small 𝛿. What is the largest such 𝛿?

  • A 𝑥 + 4 𝑥
  • B 𝑥 𝑥 + 2
  • C 𝑥 𝑥 + 4
  • D 𝑥 + 4 𝑥
  • E 𝑥 𝑥 + 4

The condition 2𝑥>12 was to ensure truth to the figure. Does the same 𝛿 work if 𝑥≥4?

  • Ano
  • Byes

It would appear that all your arguments depended upon the fact that the graph of the function 𝑓 was concave up. Let 𝑓(𝑥)=𝑒. Find 𝛿>0 so that |𝑓(𝑥)−𝑓(𝑥)|<14∗ whenever |𝑥−𝑥|<𝛿∗. Assume that 𝑥 is positive.

  • A 𝛿 = ( 4 + 𝑒 ) + ( 4 ) l n l n
  • B 𝛿 = 2 𝑥 − 4 + ( 4 + 𝑒 ) l n l n
  • C 𝛿 = ( 4 + 𝑒 ) − ( 4 ) l n l n
  • D 𝛿 = 2 𝑥 − 4 + ( 4 + 𝑒 ) l n l n
  • E 𝛿 = ( 4 + 𝑒 ) − ( 4 ) l n l n

Q6:

Sometimes, we do not have concavity to help determine an explicit 𝛿 that shows continuity at a point. The figure shows the graph of 𝑓(𝑥)=3+𝑥+𝑥𝑥>0,3+𝑥−4𝑥𝑥≤0,

together with its tangent line 𝑦=𝑥+3 at the inflection point (0,3).

What is 𝑓(𝑥) for 𝑥>0?

  • A − √ 4 𝑥 − 1 1 + 1 2
  • B √ 4 𝑥 − 1 1 − 1 2
  • C 1 + √ 4 9 − 1 6 𝑥 8
  • D 1 − √ 4 9 − 1 6 𝑥 8

What is 𝑓(𝑥) for 𝑥≤0?

  • A 1 − √ 4 9 − 1 6 𝑥 8
  • B √ 4 𝑥 − 1 1 − 1 2
  • C 1 + √ 4 9 − 1 6 𝑥 8
  • D − √ 4 𝑥 − 1 1 + 1 2

By considering the graph near (0,3), find the largest 𝛿 so that |𝑓(𝑥)−3|<𝜀 whenever |𝑥|<𝛿.

  • A 1 − √ 1 6 𝜀 + 1 8
  • B √ 4 𝜀 + 1 − 1 2
  • C √ 1 6 𝜀 + 1 − 1 8
  • D 1 − √ 4 𝜀 + 1 2

Q7:

The graph of 𝑓(𝑥)=5−(𝑥−2) is concave down and decreasing when 𝑥>2. We want to find the maximal 𝛿 so that, for a given 𝜀>0, it follows that if |𝑥−3|<𝛿 then |𝑓(𝑥)−𝑓(3)|<𝜀. This 𝛿 will, of course, be a function of 𝜀.

What is the inverse function near 𝑥=3? Give an expression for 𝑓(𝑥).

  • A 2 − √ 5 − 𝑥
  • B 2 + √ 5 − 𝑥
  • C 2 + √ 𝑥 − 5
  • D 2 + √ 5 + 𝑥
  • E 2 − √ 5 + 𝑥

Using the concavity of the graph of 𝑓, determine which point is closer to 3, 𝑓(4+𝜀) or 𝑓(4−𝜀), along the horizontal axis. (Do not evaluate them and consider only small 𝜀.)

  • A 𝑓 ( 4 + 𝜀 )
  • B 𝑓 ( 4 − 𝜀 )

From your answers above, find an expression — in terms of 𝜀 — for the largest 𝛿 so that if |𝑥−3|<𝛿 then 4−𝜀<𝑓(𝑥)<4+𝜀.

  • A 2 + √ 1 + 𝜀
  • B √ 1 + 𝜀 − 1
  • C 1 − √ 1 − 𝜀
  • D 2 + √ 1 − 𝜀

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