Calculus
Limits of 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 \[ \lim_{x \to 0} \frac{x - f(x)}{x+ f(x)}. \]