Evaluating Limits

If \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist, then:
\(\displaystyle \lim_{x \to c}\bigl( kf(x) \bigr) = k \Bigl(\lim_{x \to c} f(x)\Bigr) \,\text{ for } k \in \mathbf{R}\)
\(\displaystyle \lim_{x \to c}\bigl( f(x) \pm g(x) \bigr) = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)\)
\(\displaystyle \lim_{x \to c}\bigl( f(x) \cdot g(x) \bigr) = \Bigl(\lim_{x \to c} f(x)\Bigr) \cdot \Bigl(\lim_{x \to c} g(x)\Bigr)\)
\(\displaystyle \lim_{x \to c}\bigl( f(x) \bigr)^n = \Bigl(\lim_{x \to c} f(x) \Bigr)^n \,\text{ for } n \in \mathbf{R}\)
\(\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)} \,\text{ so long as } \lim_{x \to c} g(x) \neq 0\)

As exercise, for each of the following limits, either determine its value or decide that the limit does not exist. For any limit whose value, though it exists, proves difficult to express exactly, calculate a decimal approximation to the value accurate to within ±one-thousandth.

\(\displaystyle \lim_{x\to 0^-}\frac{x^3+x^2}{x^3+2x^2}\)
\(\displaystyle \lim_{x\to 7^-}\Bigl(|x^2+x|-x\Bigr)\)
\(\displaystyle \lim_{x\to -1^+}\sqrt{1-x^2}\)
\(\displaystyle \lim_{x\to -1^-}\sqrt{1-x^2}\)
\(\displaystyle \lim_{x\to 0^-}\frac{|x|}{x}\)
\(\displaystyle \lim_{x\to 1^-}\arcsin(x)\)
\(\displaystyle \lim_{x\to-1}\frac{1}{x-1}\)
\(\displaystyle \lim_{x\to4}\frac{1}{x-4}\)
\(\displaystyle \lim_{x\to2}\frac{1}{x-2}\)
\(\displaystyle \lim_{x\to-3}\frac{x^2-9}{x+3}\)
\(\displaystyle \lim_{x\to3}\frac{x^2-9}{x-3}\)
\(\displaystyle \lim_{x\to-1}\frac{x^2+2x+1}{x+1} \)
\(\displaystyle \lim_{x\to-1}\frac{x^3+1}{x+1} \)
\(\displaystyle \lim_{x\to4}\frac{x^2+5x-36}{x^2-16}\)
\(\displaystyle \lim_{x\to25}\frac{x-25}{\sqrt{x}-5}\)
\(\displaystyle \lim_{x\to0}\frac{|x|}{x}\)
\(\displaystyle \lim_{x\to2}\frac{1}{(x-2)^2}\)
\(\displaystyle \lim_{x\to3}\frac{\sqrt{x^2+16}}{x-3}\)
\(\displaystyle \lim_{x\to-2}\frac{3x^2-8x-3}{2x^2-18}\)
\(\displaystyle \lim_{x\to2}\frac{x^2+2x+1}{x^2-2x+1}\)
\(\displaystyle \lim_{x\to3}\frac{x+3}{x^2-9}\)
\(\displaystyle \lim_{x\to-1}\frac{x+1}{x^2+x}\)
\(\displaystyle \lim_{x\to1}\frac{1}{x^2+1}\)
\(\displaystyle \lim_{x\to1}\Bigl(x^2+5x-\frac{1}{2-x}\Bigr)\)
\(\displaystyle \lim_{x\to0}\frac{x^2}{x^2+2x-3}\)
\(\displaystyle \lim_{x\to1}\frac{x^2-1}{x^2+2x-3}\)
\(\displaystyle \lim_{x\to1}\frac{5x}{x^2+2x-3}\)
\(\displaystyle \lim_{x\to -1}\ln\Bigr(\sqrt{x^2-1}\Bigl)\)
\(\displaystyle \lim_{x\to 1}\arcsin(x)\)
\(\displaystyle \lim_{x\to\infty}\frac{-x+\pi}{x^2+3x+2}\)
\(\displaystyle \lim_{x\to-\infty}\frac{x^2+2x+1}{3x^2+1}\)
\(\displaystyle \lim_{x\to-\infty}\frac{3x^2+x}{2x^2-15}\)
\(\displaystyle \lim_{x\to-\infty}\Bigr(3x^2-2x+1\Bigr)\)
\(\displaystyle \lim_{x\to\infty}\frac{2x^2-32}{x^3-64}\)
\(\displaystyle \lim_{x\to\infty}6\)
\(\displaystyle \lim_{x\to\infty}\frac{3x^2+4x}{x^4+2}\)
\(\displaystyle \lim_{x\to-\infty}\frac{2x+3x^2+1}{2x^2+3}\)
\(\displaystyle \lim_{x\to-\infty}\frac{x^3-3x^2+1}{3x^2+x+5}\)
\(\displaystyle \lim_{x\to\infty}\frac{x^2+2}{x^3-2}\)
\(\displaystyle \lim_{x\to\infty}\frac{\sin\bigl(\frac{1}{x}\bigr)}{x}\)
\(\displaystyle \lim_{x\to-\infty}x^{2}\cos\biggl(\frac{1}{x}\biggr)\)
\(\displaystyle \lim_{x\to\infty}\frac{\sin(\arctan(x))}{\arctan(-x)}\)