The value of `lim_(xto(1)/(sqrt(2))) (x-cos(sin^(-1)x))/(1-tan(sin^(-1)x))" is "`

Options

A. `-(1)/(sqrt(2))`
B. -1
C. non-existent
D. none of these

Explanation

To solve the limit \( \lim_{x \to \frac{1}{\sqrt{2}}} \frac{x - \cos(\sin^{-1} x)}{1 - \tan(\sin^{-1} x)} \), we will follow these steps:

### Step 1: Substitute \( \theta = \sin^{-1}(x) \)
Let \( \theta = \sin^{-1}(x) \). Then, we have:
\[ x = \sin(\theta) \]
As \( x \to \frac{1}{\sqrt{2}} \), \( \theta \to \frac{\pi}{4} \).

### Step 2: Rewrite the limit in terms of \( \theta \)
Now, we can rewrite the limit:
\[ \lim_{\theta \to \frac{\pi}{4}} \frac{\sin(\theta) - \cos(\theta)}{1 - \tan(\theta)} \]

### Step 3: Express \( \tan(\theta) \)
Recall that:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
Thus, we can rewrite the denominator:
\[ 1 - \tan(\theta) = 1 - \frac{\sin(\theta)}{\cos(\theta)} = \frac{\cos(\theta) - \sin(\theta)}{\cos(\theta)} \]

### Step 4: Substitute back into the limit
Substituting this back into our limit gives:
\[ \lim_{\theta \to \frac{\pi}{4}} \frac{\sin(\theta) - \cos(\theta)}{\frac{\cos(\theta) - \sin(\theta)}{\cos(\theta)}} \]
This simplifies to:
\[ \lim_{\theta \to \frac{\pi}{4}} \frac{\sin(\theta) - \cos(\theta)}{1} \cdot \frac{\cos(\theta)}{\cos(\theta) - \sin(\theta)} \]

### Step 5: Simplify the expression
This can be simplified further:
\[ \lim_{\theta \to \frac{\pi}{4}} \cos(\theta) \cdot \frac{\sin(\theta) - \cos(\theta)}{\cos(\theta) - \sin(\theta)} = \lim_{\theta \to \frac{\pi}{4}} \cos(\theta) \cdot (-1) \]

### Step 6: Evaluate the limit
Now, we evaluate the limit as \( \theta \to \frac{\pi}{4} \):
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Thus, the limit becomes:
\[ -\frac{1}{\sqrt{2}} \]

### Final Answer
The value of the limit is:
\[ \boxed{-\frac{1}{\sqrt{2}}} \]