1. Evaluate each of the following limits. \[ \lim\limits_{x \to 2} \frac{x^7-x^6-64}{x^3-8} \qquad \lim\limits_{x \to 0} \frac{\tan(3x)}{\sin(7x)} \qquad \lim\limits_{x \to 0} \frac{\cos\!\big(5x^2\big)}{\sin\!\big(x^2\big)} \] \[ \lim\limits_{x \to 1} \tan\!\left(\frac{\pi x }{2}\right)\ln(x) \qquad \lim\limits_{x \to \infty} \ln(x)-x \qquad \lim\limits_{x \to \infty} x^2\ln\!\big(x^2\big) \qquad \lim\limits_{x \to \infty} \frac{\ln\!\big(\sqrt{x}\big)}{x^2} \] \[ \lim\limits_{x \to \infty} \big(\sin(x)\big)^{\sqrt{x}} \qquad \lim\limits_{x \to \infty} x^{1/\ln(x)} \qquad \lim\limits_{x \to 0^+} (7x+2)^{\mathrm{cot}(x)} \]
2. Evaluate this limit. \[ \lim\limits_{x \to \infty} \sqrt{x^2+x+1}-x \]