Evaluate the following limits: either determine the limit’s value or decide that the limit does not exist. These exercises are intended to be done either algebraically, simplifying the formula to remove a point-discontinuity, or computationally. Refrain from using technology to plot a graph of the expression except to check your conclusions from algebraic and computational work. 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.
\(\displaystyle \lim\limits_{x \to 4} \frac{2x-9}{x+4} \)
\(\displaystyle \lim\limits_{x \to -3} \frac{|x+3|}{2x+6} \)
\(\displaystyle \lim\limits_{x \to \tfrac{1}{4}^{-}} \frac{4x-1}{|4x^3-x^2|} \)
\(\displaystyle \lim\limits_{x \to 0^-} \left(\frac{1}{|x|} - \frac{1}{x}\right) \)
\(\displaystyle \lim\limits_{x \to 0} \frac{x}{x^2+5x} \)
\(\displaystyle \lim\limits_{x \to -4} \frac{x^2+4}{x+4} \)
\(\displaystyle \lim\limits_{x \to 2} \frac{x-2}{x^2+9x-22} \)
\(\displaystyle \lim\limits_{x \to 7} \frac{x^2-6x-7}{7-x} \)
\(\displaystyle \lim\limits_{x \to -4} \frac{3x^2+9x-5}{x^2-16} \)
\(\displaystyle \lim\limits_{x \to 100} \frac{x-100}{10-\sqrt{x}} \)
\(\displaystyle \lim\limits_{x \to 0} \left(\frac{1}{x} - \frac{1}{x^2+1}\right) \)
\(\displaystyle \lim\limits_{x \to \infty} \sqrt{x} \)
\(\displaystyle \lim\limits_{x \to 7^+} \frac{x^2-x-20}{x-7} \)
\(\displaystyle \lim\limits_{x \to 7^+} \frac{x^2-3x-28}{x-7} \)
\(\displaystyle \lim\limits_{x \to 7} \frac{49}{(x-7)^2} \)
\(\displaystyle \lim\limits_{x \to \infty} \frac{3x^2-7x+5}{2x-1} \)
\(\displaystyle \lim\limits_{x \to \infty} \frac{3x^2-7x+5}{(2x-1)^2} \)
\(\displaystyle \lim\limits_{x \to \infty} \frac{3x^2-7x+5}{(2x-1)^3} \)
\(\displaystyle \lim\limits_{x \to 1} \frac{1-\sqrt{x}}{1-x} \)
\(\displaystyle \lim\limits_{x \to 0} \frac{1-\sqrt{1-x^2}}{x} \)
\(\displaystyle \lim\limits_{x \to 0} \frac{1-\sqrt{1-x^2}}{x^2} \)
\(\displaystyle \lim\limits_{x \to 0}\frac{x^4+x^3}{x^4+3x^3}\)
\(\displaystyle \lim\limits_{x \to 3^+}\Bigl(\bigr|x^2+x\bigl|-x\Bigr)\)
\(\displaystyle \lim\limits_{x \to -1^+}\sqrt{1-x^2}\)
\(\displaystyle \lim\limits_{x \to -1^-}\sqrt{1-x^2}\)
\(\displaystyle \lim\limits_{x \to 1^-}\arcsin(x)\)
\(\displaystyle \lim\limits_{x \to 0^-}\frac{x}{|x|}\)
\(\displaystyle \lim\limits_{x \to 3}\frac{x-3}{x+3}\)
\(\displaystyle \lim\limits_{x \to -3^+}\frac{x-3}{x+3}\)
\(\displaystyle \lim\limits_{x \to -1}\frac{1}{x-1}\)
\(\displaystyle \lim\limits_{x \to 7}\frac{1}{x-7}\)
\(\displaystyle \lim\limits_{x \to 7}\frac{1}{(x-7)^4}\)
\(\displaystyle \lim\limits_{x \to -3}\frac{x^2-9}{x+3}\)
\(\displaystyle \lim\limits_{x \to 3}\frac{x^2-9}{x-3}\)
\(\displaystyle \lim\limits_{x \to -1}\frac{x^2+2x+1}{x+1} \)
\(\displaystyle \lim\limits_{x \to -1}\frac{x^3+1}{x+1} \)
\(\displaystyle \lim\limits_{x \to -1}\frac{x^3+3x-2}{(x+1)^2} \)
\(\displaystyle \lim\limits_{x \to -1}\frac{x^3-6x^2+11x-6}{(x+1)^2} \)
\(\displaystyle \lim\limits_{x \to 4}\frac{x^2+5x-36}{x^2-16}\)
\(\displaystyle \lim\limits_{x \to 25}\frac{x-25}{\sqrt{x}-5}\)
\(\displaystyle \lim\limits_{x \to 3^+}\frac{1}{(x-3)^3}\)
\(\displaystyle \lim\limits_{x \to 4}\frac{\sqrt{x^2+9}}{x-4}\)
\(\displaystyle \lim\limits_{x \to -2}\frac{3x^2-8x-3}{2x^2-18}\)
\(\displaystyle \lim\limits_{x \to 2}\frac{x^2+2x+1}{x^2-2x+1}\)
\(\displaystyle \lim\limits_{x \to 3}\frac{x+3}{x^2-9}\)
\(\displaystyle \lim\limits_{x \to -1}\frac{x+1}{x^2+x}\)
\(\displaystyle \lim\limits_{x \to 1^+}\frac{1}{x^2+1}\)
\(\displaystyle \lim\limits_{x \to 1}\Bigl(3-\tfrac{1}{3-x}-x\Bigr)\)
\(\displaystyle \lim\limits_{x \to 0}\frac{x^2}{x^2+2x-3}\)
\(\displaystyle \lim\limits_{x \to 1}\frac{x^2-1}{x^2+2x-3}\)
\(\displaystyle \lim\limits_{x \to 1}\frac{5x}{x^2+2x-3}\)
\(\displaystyle \lim\limits_{x \to 7}\frac{\sqrt{x}-\sqrt{7}}{x-7}\)
\(\displaystyle \lim\limits_{x \to 1}\frac{3-\sqrt{8+x}}{x-1}\)
\(\displaystyle \lim\limits_{x \to -2}\frac{x^2+9x+14}{\sqrt{x+11}-3}\)
\(\displaystyle \lim\limits_{x \to 5}\frac{x^2-2x-15}{\sqrt{3+x}-5}\)
\(\displaystyle \lim\limits_{x \to 2}\frac{\sqrt{x^2+1}-\sqrt{5}}{x^2-4}\)
\(\displaystyle \lim\limits_{x \to \infty}\frac{1-x}{x^2+19x-8}\)
\(\displaystyle \lim\limits_{x \to -\infty}\frac{x^2+2x+1}{3x^2+3}\)
\(\displaystyle \lim\limits_{x \to \infty}\frac{2x^2-32}{x^3-64}\)
\(\displaystyle \lim\limits_{x \to \infty} 7\)
\(\displaystyle \lim\limits_{x \to \infty}\frac{x^4+4x}{x^3-3}\)
\(\displaystyle \lim\limits_{x \to \infty}\frac{2x^5-2x^2+11}{3x^5+x-7}\)
\(\displaystyle \lim\limits_{x \to \infty}\frac{x^2+7}{x^3-7}\)