How do you find the limit of #(x / cos (-3x))# as x approaches #-oo#?

Question

Evelyn Agard · Accepted Answer

The limit does not exist

It should be clear that the denominator oscillates between #+-1#, and the demonstrator decreases without bound as #x rarr -oo#, Hence the function increases without bound as #x rarr -oo#, This can be verified with the graph of the function: graph{x/cos(0-3x) [-80, 80, -40, 40]}

Emilia Aguilar · Answer

undefined To find the limit of (x / cos (-3x)) as x approaches negative infinity, we can use the fact that the cosine function is bounded between -1 and 1. As x approaches negative infinity, the cosine of -3x will oscillate between -1 and 1. Since the numerator, x, is unbounded and the denominator, cos (-3x), is bounded, the limit of (x / cos (-3x)) as x approaches negative infinity does not exist.

Isaiah Allen · Answer

undefined To find the limit of $ \frac{x}{\cos(-3x)} $ as $ x $ approaches negative infinity, we need to consider the behavior of the function as $ x $ becomes increasingly negative. Since cosine is a periodic function with a range of $[-1, 1]$, the denominator $ \cos(-3x) $ oscillates between -1 and 1 as $ x $ approaches negative infinity. The numerator $ x $ is a linear function, which grows without bound as $ x $ becomes increasingly negative. As a result, the limit of the function $ \frac{x}{\cos(-3x)} $ as $ x $ approaches negative infinity does not exist.