Calculus

# Limits of Composite Functions

If $\lim_{x \to 4} \frac{f(x) - 5}{x-2}=1,$ find $$\displaystyle \lim_{x \to 4} f(x).$$

Given the function $f(x) = \frac{ 6x + 4}{\sqrt{x^2+7x} - x},$ what is $$\displaystyle{\lim_{ x \rightarrow -\infty } f(x)}$$?

If $$\displaystyle \lim_{x \to 0} \frac{f(x)}{x}=6,$$ what is the value of $\lim_{x \to 0} \frac{3x^2+8f(x)}{2x^2-f(x)}?$

Find the value of

$\lim_{x \to \infty} \left( \sqrt{x^2 + \lfloor x \rfloor} - x \right).$

Assumptions and Details:
$$\lfloor \cdot \rfloor$$ is the floor function such that $$\lfloor x \rfloor = n$$ if $$n \leq x < n+1,$$ where $$n$$ is an integer.

If $$\displaystyle \lim_{x \to -1} \frac{ f(x+1) }{x+1} = 2,$$ find the value of $\lim_{x \to 0} \frac{x - f(x)}{x+ f(x)}.$

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