Limits of functions - Problem solving

         

Find the value of

limx2x+sinxx+cosx. \lim_{x \to \infty} \frac{2x + \sin x}{x + \cos x}.

Suppose a polynomial function f(x)f(x) satisfies limx1f(x)+2x1=3. \lim_{x \to 1} \frac{f(x)+2}{x-1} = 3. Find the value of limx1[f(x)]24x21. \lim_{x \to -1} \frac{[f(-x)]^2 - 4}{x^2 -1}.

Evaluate limx36x2+2613x12\displaystyle \lim_{x \to -\infty} \frac{\sqrt{36 x^2 + 26}}{13 - \frac{x}{12}}.

Consider the function f(x)=ax3+bx2+10x20x21.f(x) = \displaystyle \frac{ax^3+bx^2+10x-20}{x^2-1}. If limxf(x)=10,\displaystyle \lim_{x \to \infty}f(x)=10, what is the value of limx1f(x)?\displaystyle \lim_{x \to 1}f(x)?

For the function f(x)=x(x+2)x(x+1), f(x) = \frac{ |x(x+2)| }{ x(x+1) }, find the value of limx2+f(x)+limx0+f(x). \lim_{x \to -2^+} f(x) + \lim_{x \to 0^+} f(x).

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