The value of limx →1((x^3+2 x^2+x+1)/(x^2+2 x+3))^(1-cos(x-1))/((x-1)^2) is (A) e (B) e^1 / 2 (C

step 1 In this question, we need to evaluate the limit, that is limit extends to 1 over the expression

The value of limx →1((x^3+2 x^2+x+1)/(x^2+2...

The value of $\lim _{\mathrm{x} \rightarrow 1}\left(\frac{\mathrm{x}^{3}+2 \mathrm{x}^{2}+\mathrm{x}+1}{\mathrm{x}^{2}+2 \mathrm{x}+3}\right)^{\frac{1-\cos (\mathrm{x}-1)}{(\mathrm{x}-1)^{2}}}$ is
(A) e
(B) $\mathrm{e}^{1 / 2}$
(C) 1
(D) none of these

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Key Concepts

Limits

Limits study the behavior of functions as the input approaches a specific value. They provide a way to understand the output of functions even at points where direct evaluation might be impossible due to undefined expressions or indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when substituting a value into a limit expression yields an ambiguous form, such as 1^? or 0^0. Resolving these forms often requires advanced techniques to determine the true limit value.

Exponential Limits

Exponential limits involve expressions where a function is raised to another function that approaches 0, infinity, or another critical value. These problems are typically handled by rewriting the expression in an exponential form using logarithms to extract and evaluate the exponent.

Taylor Series Expansion

The Taylor (or Maclaurin) series provides a polynomial approximation of functions near a specific point. This approximation is particularly useful in limit problems to simplify complex expressions, such as approximating trigonometric functions for small angles.

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