Limits of Functions

Limits of Composite Functions


If limx4f(x)5x2=1, \lim_{x \to 4} \frac{f(x) - 5}{x-2}=1, find limx4f(x). \displaystyle \lim_{x \to 4} f(x).

Given the function f(x)=6x+4x2+7xx, f(x) = \frac{ 6x + 4}{\sqrt{x^2+7x} - x}, what is limxf(x)\displaystyle{\lim_{ x \rightarrow -\infty } f(x)} ?

If limx0f(x)x=6,\displaystyle \lim_{x \to 0} \frac{f(x)}{x}=6, what is the value of limx03x2+8f(x)2x2f(x)?\lim_{x \to 0} \frac{3x^2+8f(x)}{2x^2-f(x)}?

Find the value of

limx(x2+xx). \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 x=n \lfloor x \rfloor = n if nx<n+1, n \leq x < n+1, where n n is an integer.

If limx1f(x+1)x+1=2, \displaystyle \lim_{x \to -1} \frac{ f(x+1) }{x+1} = 2, find the value of limx0xf(x)x+f(x). \lim_{x \to 0} \frac{x - f(x)}{x+ f(x)}.


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