Worksheet: Comparing Rate of Growth of Functions

In this worksheet, we will practice using limits to compare the relative magnitudes of functions and their rates of change.

Q1:

For the functions 𝑓 ( π‘₯ ) = 2 π‘₯  and 𝑔 ( π‘₯ ) = 4 𝑒   , evaluate l i m  β†’ ∞ ο€½ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ )  using l’HΓ΄pital’s rule.

Q2:

Evaluate l i m  β†’ ∞   ο€Ύ π‘₯ 𝑒  using l’HΓ΄pital’s rule.

From the answer to the limit, what can you say about the growth rate of π‘₯  compared to 𝑒  as π‘₯ β†’ ∞ ?

  • AThe growth rate of π‘₯  becomes ∞ , while the growth rate of 𝑒  becomes 0 as π‘₯ β†’ ∞ .
  • BThe growth rate of π‘₯  is greater than that of 𝑒  as π‘₯ β†’ ∞ .
  • CThe growth rate of π‘₯  becomes 0, while the growth rate of 𝑒  becomes ∞ as π‘₯ β†’ ∞ .
  • DThe growth rate of π‘₯  is equal to that of 𝑒  as π‘₯ β†’ ∞ .
  • EThe growth rate of π‘₯  is smaller than that of 𝑒  as π‘₯ β†’ ∞ .

Q3:

If l i m  β†’ ∞ ο€½ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ )  = 0 , what do you notice about the growth rate of 𝑓 ( π‘₯ ) compared to 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ ?

  • AThe growth rate of 𝑓 ( π‘₯ ) becomes ∞ , while the growth rate of 𝑔 ( π‘₯ ) becomes 0 as π‘₯ β†’ ∞ .
  • BThe growth rate of 𝑓 ( π‘₯ ) is equal to that of 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ .
  • CThe growth rate of 𝑓 ( π‘₯ ) is greater than that of 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ .
  • DThe growth rate of 𝑓 ( π‘₯ ) becomes 0, while the growth rate of 𝑔 ( π‘₯ ) becomes ∞ as π‘₯ β†’ ∞ .
  • EThe growth rate of 𝑓 ( π‘₯ ) is smaller than that of 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ .

Q4:

If l i m  β†’ ∞ ο€½ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ )  = ∞ , what do you notice about the growth rate of 𝑓 ( π‘₯ ) compared to 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ ?

  • AThe growth rate of 𝑓 ( π‘₯ ) is greater than that of 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ .
  • BThe growth rate of 𝑓 ( π‘₯ ) becomes ∞ , while the growth rate of 𝑔 ( π‘₯ ) becomes 0 as π‘₯ β†’ ∞ .
  • CThe growth rate of 𝑓 ( π‘₯ ) is equal to that of 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ .
  • DThe growth rate of 𝑓 ( π‘₯ ) is smaller than that of 𝑔 ( π‘₯ ) as π‘₯ β†’ ∞ .
  • EThe growth rate of 𝑓 ( π‘₯ ) becomes 0, while the growth rate of 𝑔 ( π‘₯ ) becomes ∞ as π‘₯ β†’ ∞ .

Q5:

Compare the growth rate of the two functions 𝑓 ( π‘₯ ) = ( π‘₯ + 4 )  and 𝑔 ( π‘₯ ) = π‘₯ l n using limits as π‘₯ β†’ ∞ .

  • AThe growth rate of 𝑓 ( π‘₯ ) is smaller than the growth rate of 𝑔 ( π‘₯ ) .
  • BThe growth rate of 𝑓 ( π‘₯ ) becomes ∞ , while the growth rate of 𝑔 ( π‘₯ ) becomes 0 as π‘₯ β†’ ∞ .
  • CThe growth rate of 𝑓 ( π‘₯ ) is greater than the growth rate of 𝑔 ( π‘₯ ) .
  • DThe growth rate of 𝑓 ( π‘₯ ) becomes 0, while the growth rate of 𝑔 ( π‘₯ ) becomes ∞ as π‘₯ β†’ ∞ .
  • EThe growth rate of 𝑓 ( π‘₯ ) is equal to the growth rate of 𝑔 ( π‘₯ ) .

Q6:

For the functions 𝑓 ( π‘₯ ) = ( 2 π‘₯ ) l n and 𝑔 ( π‘₯ ) = 4 π‘₯  , evaluate l i m  β†’ ∞ ο€½ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ )  using l’HΓ΄pital’s rule.

Q7:

Compare the growth rate of the two functions 𝑓 ( π‘₯ ) = π‘₯   and 𝑔 ( π‘₯ ) = 1 0  using limits as π‘₯ β†’ ∞ .

  • AThe growth rate of 𝑓 ( π‘₯ ) becomes 0, while the growth rate of 𝑔 ( π‘₯ ) becomes ∞ as π‘₯ β†’ ∞ .
  • BThe growth rate of 𝑓 ( π‘₯ ) is equal to the growth rate of 𝑔 ( π‘₯ ) .
  • CThe growth rate of 𝑓 ( π‘₯ ) is greater than the growth rate of 𝑔 ( π‘₯ ) .
  • DThe growth rate of 𝑔 ( π‘₯ ) becomes 0, while the growth rate of 𝑓 ( π‘₯ ) becomes ∞ as π‘₯ β†’ ∞ .
  • EThe growth rate of 𝑔 ( π‘₯ ) is greater than the growth rate of 𝑓 ( π‘₯ ) .

Q8:

Compare the growth rate of the two functions 𝑓 ( π‘₯ ) = 𝑒  and 𝑔 ( π‘₯ ) = π‘₯ l n using limits as π‘₯ β†’ ∞ .

  • AThe growth rate of 𝑒  becomes 0, while the growth rate of l n π‘₯ becomes ∞ as π‘₯ β†’ ∞ .
  • BThe growth rate of l n π‘₯ is greater than the growth rate of 𝑒  .
  • CThe growth rate of l n π‘₯ becomes 0, while the growth rate of 𝑒  becomes ∞ as π‘₯ β†’ ∞ .
  • DThe growth rate of 𝑒  is greater than the growth rate of l n π‘₯ .
  • EThe growth rate of 𝑒  is equal to the growth rate of l n π‘₯ .

Q9:

Compare the growth rate of the two functions 𝑓 ( π‘₯ ) = π‘₯  and 𝑔 ( π‘₯ ) = 1 π‘₯ using limits as π‘₯ β†’ ∞ .

  • AThe growth rate of 𝑓 ( π‘₯ ) is equal to the growth rate of 𝑔 ( π‘₯ ) .
  • BThe growth rate of 𝑓 ( π‘₯ ) is greater than the growth rate of 𝑔 ( π‘₯ ) .
  • CThe growth rate of 𝑓 ( π‘₯ ) becomes 0, while the growth rate of 𝑔 ( π‘₯ ) becomes ∞ as π‘₯ β†’ ∞ .
  • DThe growth rate of 𝑓 ( π‘₯ ) is smaller than the growth rate of 𝑔 ( π‘₯ ) .
  • EThe growth rate of 𝑓 ( π‘₯ ) becomes ∞ , while the growth rate of 𝑔 ( π‘₯ ) becomes 0 as π‘₯ β†’ ∞ .

Q10:

Consider the functions 𝑓 ( π‘₯ ) = π‘₯   and 𝑔 ( π‘₯ ) = 2  .

Evaluate l i m  β†’ ∞ ο€½ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ )  using l’HΓ΄pital’s rule.

Evaluate l i m  β†’ ∞ ο€½ 𝑔 ( π‘₯ ) 𝑓 ( π‘₯ )  using l’HΓ΄pital’s rule.

  • A1
  • B0
  • C ∞
  • D2
  • EThe limit does not exist.

What do you get from the two results about the growth rates of π‘₯   and 2  as π‘₯ β†’ ∞ ?

  • AThe growth rate of 2  is greater than that of π‘₯   as π‘₯ β†’ ∞ .
  • BThe growth rate is the same for both as π‘₯ β†’ ∞ .
  • CThe growth rate of 2  becomes 0, while the growth rate of π‘₯   becomes ∞ as π‘₯ β†’ ∞ .
  • DThe growth rate of π‘₯   becomes 0, while the growth rate of 2  becomes ∞ as π‘₯ β†’ ∞ .
  • EThe growth rate of π‘₯   is greater than that of 2  as π‘₯ β†’ ∞ .

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