Limits practice question 16
Evaluate the limit.
The term keeps oscillating and never settles. How big can it ever be, next to when is huge?
Divide the top and the bottom by . What happens to as ?
stays between and , so for .
A fraction or a decimal.
Theory UBC Math 100 §2.1.2CLP-1 §1.5Key results
Worked solution
Strategy
Divide by the dominant power, x. The cosine has no limit, but it is bounded, and a bounded quantity divided by something growing without bound tends to 0.
Working
Divide every term by x.
The cosine is at most 1 in size and 1/x tends to 0, so cos(2x)/x tends to 0.
Every small term vanishes.
Answer
The answer is five halves.
Check
At x = 1000 the quotient lies between 4999/2001 ≈ 2.498 and 5001/2001 ≈ 2.499 whatever the cosine does, close to 2.5.
Where this goes wrong
Deciding that the limit does not exist because cos(2x) oscillates. It does oscillate, but it is never more than 1 in size, and divided by x it shrinks to 0.
The idea behind it
The limits lesson, a figure to move, in a few steps
Leans on
Behaviour of 1/x as x grows, from Functions, graphs and inverses
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