Let `f(x)={{:(5x-4",",0ltxle1),(4x^(3)-3x",",1ltxlt2.):}`
Find `lim_(xrarr1)f(x).`

Explanation

To find the limit of the function \( f(x) \) as \( x \) approaches 1, we need to evaluate the left-hand limit and the right-hand limit separately.

### Step 1: Identify the function definitions
The function \( f(x) \) is defined as:
- \( f(x) = 5x - 4 \) for \( 0 < x \leq 1 \)
- \( f(x) = 4x^3 - 3x \) for \( 1 < x < 2 \)

### Step 2: Calculate the right-hand limit as \( x \) approaches 1
To find the right-hand limit \( \lim_{x \to 1^+} f(x) \):
- We use the definition of \( f(x) \) for \( x > 1 \):
\[ f(x) = 4x^3 - 3x \]
Substituting \( x = 1 \):
\[ \lim_{x \to 1^+} f(x) = 4(1)^3 - 3(1) = 4 - 3 = 1 \]

### Step 3: Calculate the left-hand limit as \( x \) approaches 1
To find the left-hand limit \( \lim_{x \to 1^-} f(x) \):
- We use the definition of \( f(x) \) for \( x \leq 1 \):
\[ f(x) = 5x - 4 \]
Substituting \( x = 1 \):
\[ \lim_{x \to 1^-} f(x) = 5(1) - 4 = 5 - 4 = 1 \]

### Step 4: Compare the left-hand and right-hand limits
Now we compare the two limits:
\[ \lim_{x \to 1^+} f(x) = 1 \]
\[ \lim_{x \to 1^-} f(x) = 1 \]

Since both the left-hand limit and the right-hand limit are equal, we can conclude that:
\[ \lim_{x \to 1} f(x) = 1 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 1} f(x) = 1 \]

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