Question

What is ( lim_{{x to 0}} {{cos x}}/{{x}} )?

A. -1
B. DNE
C. 1
D. 0
E. ( infty )

Answer 1

The limit of cos x / x as x approaches zero is negative infinity.

Step-by-step explanation:

The limit of cos x / x as x approaches zero can be found using L'Hopital's rule. L'Hopital's rule states that if the limit of the ratio of two functions is indeterminate (such as 0/0 or ∞/∞), then the limit can be found by taking the derivative of the numerator and denominator and evaluating the limit again.

In this case, we have:

limx→0 (cos x / x) = limx→0 (-sin x / 1)

Since the limit remains indeterminate, we can apply L'Hopital's rule again:

limx→0 (-sin x / 1) = limx→0 (-cos x / 0)

Now, we can evaluate the limit:

limx→0 (-cos x / 0) = (-1 / 0) = -∞

So the answer is option E: (∞). The limit of cos x / x as x approaches zero is negative infinity.

Answer 2

B. DNE

Step-by-step explanation:

Given:

`\lim_(x \to 0) (\cos(x))/(x)`

To evaluate the limit above, we need to consider the behavior of the function as x approaches 0. However, this limit does not exist in the conventional sense because the expression becomes the form:

`\Longrightarrow \lim_(x \to 0) (\cos(0))/(0) = (1)/(0)`

when x = 0, leading to a division by zero. In calculus, this is recognized as an undefined or indeterminate form.

So (B) is correct.