If f(x)=0 be a quadratic equation such that f(-π)=f(π)=0 and f((π)/(2))=-(3 π^2)/(4), then limx

step 1 In this question, it is given that F of x equals to 0 is a quadratic equation, that means F of x

If f(x)=0 be a quadratic equation such that f(-π)=f(π)=0 and...

If $f(x)=0$ be a quadratic equation such that $f(-\pi)=f(\pi)=0$ and $f\left(\frac{\pi}{2}\right)=-\frac{3 \pi^{2}}{4}$, then $\lim _{x \rightarrow-\pi} \frac{f(x)}{\sin (\sin x)}$ is equal to
(a) 0
(b) $\pi$
(c) $2 \pi$
(d) None of these

Key Concepts

Trigonometric Function Approximations

Trigonometric functions can be approximated by their Taylor series near specific points, typically around zero. This approximation is useful in evaluating limits where the argument of a trigonometric function is small, allowing sine and cosine functions to be approximated by their leading linear or quadratic terms.

Limits and Indeterminate Forms

The evaluation of limits, especially when both the numerator and denominator tend to zero, often leads to an indeterminate form of type 0/0. In such cases, techniques like L’Hôpital’s Rule or series approximations are applied to resolve the indeterminacy and determine the limit.

Factorization and Roots

For any quadratic function, knowledge of its zeros (roots) enables one to write it in the factored form a(x - r?)(x - r?), where r? and r? are the roots. This factorization is crucial for solving equations where the function equals zero and for understanding the behavior of the function near its zeros.

Quadratic Functions

A quadratic function is a second?degree polynomial usually written in the form ax² + bx + c. Such functions can be completely characterized by their coefficients and have a parabolic graph. Understanding quadratic functions includes techniques such as factoring, completing the square, and using the quadratic formula to find the roots or zeros of the function.

Determining Coefficients from Conditions

When additional information is provided about a function’s value at a particular point, one can use this condition to determine the unknown coefficients of the function. This method involves substituting the given point into the general form of the function and solving for the coefficient(s), ensuring that the function satisfies all the provided conditions.

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