91Ó°ÊÓ

Evaluate the following limits. $$\lim _{x \rightarrow \pi / 2} \frac{\sin x-1}{\sqrt{\sin x}-1}$$

Useful factorization formula Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\) where \(n\) is a positive integer and a is a real number. $$\lim _{x \rightarrow 2} \frac{x^{5}-32}{x-2}$$

Evaluate the following limits. $$\lim _{\theta \rightarrow 0} \frac{\frac{1}{2+\sin \theta}-\frac{1}{2}}{\sin \theta}$$

If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The population of a bacteria culture is given by \(p(t)=\frac{2500}{t+1}\).

Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}$$

If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The population of a culture of tumor cells is given by \(p(t)=\frac{3500 t}{t+1}\).

Useful factorization formula Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\) where \(n\) is a positive integer and a is a real number. $$\lim _{x \rightarrow-1} \frac{x^{7}+1}{x+1}$$ (Hint: Use the formula for \(\left.x^{7}-a^{7} \text { with } a=-1 .\right)\)

Evaluate the following limits. $$\lim _{x \rightarrow 0^{+}} \frac{1-\cos ^{2} x}{\sin x}$$

If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The amount of drug (in milligrams) in the blood after an IV tube is inserted is \(m(t)=200\left(1-2^{-t}\right)\).

Useful factorization formula Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\) where \(n\) is a positive integer and a is a real number. $$\lim _{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}$$