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Magazine Problem 67
\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. $$\text { Find } \lim _{x \rightarrow 4^{-}} x \sqrt{16-x^{2}}$$
Problem 68
In Exercises \(65-72, \lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as \(x\) approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Explain why \(\lim _{x \rightarrow 0^{+}} 2^{1 / x} \neq \lim _{x \rightarrow 0^{-}} 2^{1 / x}.\)
Problem 68
\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. $$\text { Explain why } \lim _{x \rightarrow 0^{+}} 2^{1 / x} \neq \lim _{x \rightarrow 0^{-}} 2^{1 / x}$$
Problem 69
In Exercises \(65-72, \lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as \(x\) approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. For \(f(x)=\frac{x}{|x|},\) find \(\lim _{x \rightarrow 0^{-}} f(x)\) and \(\lim _{x \rightarrow 0^{+}} f(x) .\) Is \(f(x)\) contin- uous at \(x=0 ?\) Explain.
Problem 69
\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. For \(f(x)=\frac{x}{|x|},\) find \(\lim _{x \rightarrow 0^{-}} f(x)\) and \(\lim _{x \rightarrow 0^{+}} f(x) .\) Is \(f(x)\) continuous at \(x=0 ?\) Explain.
Problem 70
\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. In Einstein's theory of relativity, the length \(L\) of an object moving at a velocity \(v\) is \(L=L_{0} \sqrt{1-\frac{v^{2}}{c^{2}}},\) where \(c\) is the speed of light and \(L_{0}\) is the length of the object at rest. Find lim \(L\) and explain why a limit from the left is used.
Problem 71
In Exercises \(65-72, \lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as \(x\) approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2} ?\)
Problem 71
\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2} ?\)
Problem 72
\(\lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as x approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x-2}}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{\sqrt{x-2}} ?\)
Problem 72
In Exercises \(65-72, \lim _{x \rightarrow a^{-}} f(x)\) means to find the limit as \(x\) approaches a from the left only, and \(\lim _{x \rightarrow a^{+}} f(x)\) means to find the limit as \(x\) approaches a from the right only. These are called one-sided limits. Solve the following problems. Is there a difference between \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x-2}}\) and \(\lim _{x \rightarrow 2^{+}} \frac{1}{\sqrt{x-2}} ?\)
