If the function f(x) satisfies \(\lim_{x\rightarrow 1}\) \(\frac{f(x)-2}{x^2-1}\) =\(\pi\), evaluate \(\lim_{x\rightarrow 1}\) f(x).

Question

Let \( f(x) = \int \frac{(2 - x^2) \cdot e^x \cdot \sqrt{1 + x} \cdot (1 - x)^{3/2}}{dx} \). If \( f(0) = 0 \), then \( f\left(\frac{1}{2}\right) \) is equal to:

Answer 1

Given:

lim_x→1 ( f(x) − 2 ) / ( x² − 1 ) = π

Step 1: Examine the denominator

As x → 1,

x² − 1 = (x − 1)(x + 1) → 0

Step 2: Apply the condition for existence of the limit

For the limit

lim_x→1 ( f(x) − 2 ) / ( x² − 1 )

to exist and be finite, the numerator must also tend to zero.

Therefore,

lim_x→1 [ f(x) − 2 ] = 0

Step 3: Evaluate the required limit

lim_x→1 f(x) − 2 = 0

⇒

lim_x→1 f(x) = 2

Final Answer:

lim_x→1 f(x) = 2

If the function f(x) satisfies \(\lim_{x\rightarrow 1}\) \(\frac{f(x)-2}{x^2-1}\) =\(\pi\), evaluate \(\lim_{x\rightarrow 1}\) f(x).

Solution and Explanation

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