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Worksheet: Limits of Oscillating Functions

Q1:

Investigate the behavior of 𝑓 ( π‘₯ ) = 2 ο€Ό 1 π‘₯  c o s as π‘₯ tends to 0.

Complete the table of values 𝑓 ( π‘₯ ) for values of π‘₯ that get closer to zero.

π‘₯ 1 9 9 πœ‹ 1 1 0 0 πœ‹ 1 9 9 9 πœ‹ 1 1 0 0 0 πœ‹ 1 9 9 9 9 πœ‹ 1 1 0 0 0 0 πœ‹
𝑓 ( π‘₯ )
  • Aβˆ’1, 1, βˆ’1, 1, βˆ’1, 1
  • B2, 2, 2, 2, 2, 2
  • C1, 1, 1, 1, 1, 1
  • Dβˆ’2, 2, βˆ’2, 2, βˆ’2, 2

What does this suggest about the graph of 𝑓 close to zero?

  • AThat it oscillates rapidly between βˆ’2 and 2
  • BThat it changes randomly
  • CThat it decreases without bound
  • DThat it increases without bound
  • EThat it approaches 2

Hence, evaluate l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) .

  • A βˆ’ ∞
  • B βˆ’ 2
  • CThe limit does not exist.
  • D2
  • E ∞

Q2:

The function 𝑓 ( π‘₯ ) = ( π‘₯ ) s i n oscillates between βˆ’1 and 1.

Evaluate l i m s i n  β†’  ( π‘₯ ) , if the limit exists.

Evaluate l i m s i n  β†’    ο€Ή π‘₯  , if the limit exists.

  • AThe limit does not exist.
  • B0
  • Cβˆ’1
  • D1
  • E ∞

Evaluate l i m s i n  β†’  ( π‘₯ ) π‘₯ , if the limit exists.