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Lesson Worksheet: Formal Definition of Infinite Limits and Limits at Infinity Mathematics

In this worksheet, we will practice interpreting the formal epsilon–delta definition of limits at infinity and limits that tend to infinity as x approaches a certain value.


As we prove that lim1𝑥=0, what should be the relation between 𝑀 and 𝜖 such that |||1𝑥0|||<𝜖 and 𝑥>𝑀?

  • A𝑀=2𝜖
  • B𝑀=2𝜖
  • C𝑀=4𝜖
  • D𝑀=1𝜖
  • E𝑀=1𝜖


Suppose that lim116(𝑥+1)=. Using the formal definition of limits, find the interval of 𝑥 centered around 1 so that the inequality 116(𝑥+1)<4,096 is satisfied.

  • A3332,3132
  • B98,78
  • C257256,255256
  • D6564,6364
  • E1716,1516


Find 𝛿 as a function of 𝑀 for a proof that limsec(𝑥)=.

  • A𝛿=41𝑀cos
  • B𝛿=21𝑀sin
  • C𝛿=1𝑀sin
  • D𝛿=1𝑀cos
  • E𝛿=21𝑀cos


Given that lim1𝑥=, find the value of 𝛿 such that for any 0<𝑥<𝛿, 1𝑥>𝑀, with 𝑀=625.

  • A𝛿=150
  • B𝛿=1250
  • C𝛿=225
  • D𝛿=125
  • E𝛿=1625


Find 𝛿 as a function of 𝑀 to prove that lim16𝑥3=.

  • A𝛿=16𝑀
  • B𝛿=3𝑀
  • C𝛿=16𝑀
  • D𝛿=16𝑀+3
  • E𝛿=116𝑀

This lesson includes 45 additional question variations for subscribers.

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