If `f(x)={{:((x)/(sinx)",",x gt0),(2-x",",xle0):}andg(x)={{:(x+3",",xlt1),(x^(2)-2x-2",",1lexlt2),(x-5",",xge2):}`
Then the value of `lim_(xrarr0) g(f(x))`

Options

A. is `-2`
B. is `-3`
C. is 1
D. does not exist

Explanation

To solve the problem, we need to find the limit of the composite function \( g(f(x)) \) as \( x \) approaches 0. Let's break this down step by step.

### Step 1: Determine \( f(0) \)
Since we are interested in the limit as \( x \) approaches 0, we first need to evaluate \( f(x) \) at \( x = 0 \) from the left side (i.e., \( x \to 0^- \)).
- For \( x \leq 0 \), the function \( f(x) \) is defined as:
\[ f(x) = 2 - x \]
- Therefore, as \( x \to 0^- \):
\[ f(0^-) = 2 - 0 = 2 \]

### Step 2: Evaluate \( g(f(0^-)) \)
Next, we need to find \( g(f(0^-)) \), which is \( g(2) \).
- For \( x \geq 2 \), the function \( g(x) \) is defined as:
\[ g(x) = x - 5 \]
- Thus, substituting \( x = 2 \):
\[ g(2) = 2 - 5 = -3 \]

### Step 3: Conclusion
Now, we can conclude that:
\[ \lim_{x \to 0^-} g(f(x)) = g(f(0^-)) = g(2) = -3 \]

### Final Answer
The value of \( \lim_{x \to 0} g(f(x)) \) is \( -3 \).