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Lesson Worksheet: Formal Definition of a Limit Mathematics

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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.

  • A53
  • B1255
  • C533
  • D253
  • E4710

Q2:

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

  • A55𝜀+1
  • B25𝜀+105𝜀+1
  • C25𝜀5𝜀+1
  • D−25𝜀+105𝜀+1
  • E−25𝜀5𝜀+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.

  • A𝑎𝜀+2𝑎𝑎𝜀+1
  • B𝑎𝜀−1𝑎𝜀
  • C𝑎𝜀+1𝑎𝜀
  • D𝑎𝜀𝑎𝜀+1
  • E𝑎𝜀−2𝑎𝑎𝜀+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?

  • A𝑏
  • B𝑎

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 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).

  • A2−√5−𝑥
  • B2+√5−𝑥
  • C2+√𝑥−5
  • D2+√5+𝑥
  • E2−√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.)

  • A𝑓(4+𝜀)
  • B𝑓(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.

  • A2+√1+𝜀
  • B√1+𝜀−1
  • C1−√1−𝜀
  • D2+√1−𝜀

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−√4𝑥−11+12
  • B√4𝑥−11−12
  • C1+√49−16𝑥8
  • D1−√49−16𝑥8

What is 𝑓(𝑥) for 𝑥≤0?

  • A1−√49−16𝑥8
  • B√4𝑥−11−12
  • C1+√49−16𝑥8
  • D−√4𝑥−11+12

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

  • A1−√16𝜀+18
  • B√4𝜀+1−12
  • C√16𝜀+1−18
  • D1−√4𝜀+12

Q7:

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?

  • A𝑏
  • B𝑎

What is 𝑎 in terms of 𝑥?

  • A4+𝑥2𝑥
  • B2𝑥4+𝑥
  • C4+𝑥4𝑥
  • D4𝑥
  • E4𝑥4+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?

  • A𝑥+4𝑥
  • B𝑥𝑥+2
  • C𝑥𝑥+4
  • D𝑥+4𝑥
  • E𝑥𝑥+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.

  • A𝛿=(4+𝑒)+(4)lnln
  • B𝛿=2𝑥−4+(4+𝑒)lnln
  • C𝛿=(4+𝑒)−(4)lnln
  • D𝛿=2𝑥−4+(4+𝑒)lnln
  • E𝛿=(4+𝑒)−(4)lnln

Q8:

The graph of 𝑓(𝑥)=11+𝑥 is concave down and decreasing when 0<𝑥<1√3. By considering which of 𝑓45−𝜀 and 𝑓45+𝜀 is necessarily closer to 12 on the 𝑥-axis, find an expression for the largest 𝛿 in terms of 𝜀 so that if |||𝑥−12|||<𝛿, then 45−𝜀<11+𝑥<45+𝜀.

  • A1−5𝜀4+5𝜀+12
  • B1−5𝜀4−5𝜀−12
  • C1+5𝜀4−5𝜀−12
  • D1−5𝜀4+5𝜀−12
  • E1+5𝜀4−5𝜀+12

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