lim⁡𝑥→0cos⁡(𝑥)−𝑥x→0lim​ cos(x)−x =A.Does not existB.1221​ C.0D.1E.3223​ ​ SUBMITarrow_backPREVIOUS

To solve the limit $\lim_{x \to 0} \frac{\cos(x) - x}{x}$, we will follow these steps:

1. Break Down the Problem

We need to evaluate the limit: $\lim_{x \to 0} \frac{\cos(x) - x}{x}$ This involves determining the behavior of the numerator and the denominator as $x$ approaches 0.

2. Relevant Concepts

We can apply L'Hôpital's Rule because both the numerator $\cos(x) - x$ and the denominator $x$ approach 0 as $x$ approaches 0.

3. Analysis and Detail

Step 1: Direct Substitution

Substituting $x = 0$ into $\cos(x)$: $\cos(0) = 1$ Thus, the expression $\frac{\cos(0) - 0}{0}$ simplifies to $\frac{1 - 0}{0}$ which is $\frac{1}{0}$, leading us to an indeterminate form $\frac{0}{0}$.

Step 2: Apply L'Hôpital's Rule

We differentiate the numerator and the denominator:

• The derivative of the numerator $\cos(x) - x$ is $-\sin(x) - 1$.
• The derivative of the denominator $x$ is $1$.

Now, compute the limit again: $\lim_{x \to 0} \frac{-\sin(x) - 1}{1}$

Step 3: Evaluate the Limit

Substituting $x = 0$: $\lim_{x \to 0} (-\sin(0) - 1) = -0 - 1 = -1$

4. Verify and Summarize

We conclude our calculations, verifying that: $\lim_{x \to 0} \frac{\cos(x) - x}{x} = -1$

Final Answer

Thus, the limit is $-1$, which means the correct option for this limit does not exist in the given choices. Hence, the answer is not listed among A, B, C, D, and E.