91Ó°ÊÓ

Compute the limits. $$ \lim _{x \rightarrow 0} \frac{2+(1 / x)}{3-(2 / x)} $$

Compute the limits. $$ \lim _{x \rightarrow 0+} \frac{1+5 / \sqrt{x}}{2+1 / \sqrt{x}} $$

Suppose a car is driving north along a road at \(80 \mathrm{~km} / \mathrm{hr}\) and an airplane is flying east at speed \(200 \mathrm{~km} / \mathrm{hr}\). Their paths crossed at a point \(P\). At a certain time, the car is 10 kilometers north of \(P\) and the airplane is 15 kilometers to the east of \(P\) at an altitude of 2 \(\mathrm{km}\) -gaining altitude at \(10 \mathrm{~km} / \mathrm{hr}\). How fast is the distance between car and airplane changing?

Compute the limits. $$ \lim _{x \rightarrow 0+} \frac{3+x^{-1 / 2}+x^{-1}}{2+4 x^{-1 / 2}} $$

A light shines from the top of a pole \(20 \mathrm{~m}\) high. An object is dropped from the same height from a point \(10 \mathrm{~m}\) away, so that its height at time \(t\) seconds is \(h(t)=20-9.8 t^{2} / 2\). How fast is the object's shadow moving on the ground one second later?

Compute the limits. $$ \lim _{x \rightarrow \infty} \frac{x+x^{1 / 2}+x^{1 / 3}}{x^{2 / 3}+x^{1 / 4}} $$

Compute the limits. $$ \lim _{t \rightarrow \infty} \frac{1-\sqrt{\frac{t}{t+1}}}{2-\sqrt{\frac{4 t+1}{t+2}}} $$

Compute the limits. $$ \lim _{t \rightarrow \infty} \frac{1-\frac{t}{t-1}}{1-\sqrt{\frac{t}{t-1}}} $$

Compute the limits. $$ \lim _{x \rightarrow-\infty} \frac{x+x^{-1}}{1+\sqrt{1-x}} $$

Compute the limits. $$ \lim _{x \rightarrow-\infty} \frac{x+x^{-1}}{1+\sqrt{1-x}} $$