Question

Accept on faith that the following familiar functions are continuous on their domains: sin \(x\), cos \(x\), \(e^x\), \(2^x\), log_e \(x\) for \(x > 0\), \(x^p\) for \(x > 0\) [\(p\) any real number]. Use these facts and theorems in this section to prove the following functions are also continuous. (a) \(\log_e(1 + \cos^4 x)\) (b) \([\sin^2 x + \cos^6 x]^\pi\) (c) \(2^{2^x}\) (d) \(8^x\) (e) tan \(x\) for \(x \neq\) odd multiple of \(\frac{\pi}{2}\) (f) \(x \sin(\frac{1}{x})\) for \(x \neq 0\) (g) \(x^2 \sin(\frac{1}{x})\) for \(x \neq 0\) (h) \(\frac{1}{x} \sin(\frac{1}{x^2})\) for \(x \neq 0\)

          Accept on faith that the following familiar functions are continuous
on their domains: sin \(x\), cos \(x\), \(e^x\), \(2^x\), log_e \(x\) for \(x > 0\), \(x^p\) for \(x > 0\)
[\(p\) any real number]. Use these facts and theorems in this section to
prove the following functions are also continuous.
(a) \(\log_e(1 + \cos^4 x)\)
(b) \([\sin^2 x + \cos^6 x]^\pi\)
(c) \(2^{2^x}\)
(d) \(8^x\)
(e) tan \(x\) for \(x \neq\) odd multiple of \(\frac{\pi}{2}\)
(f) \(x \sin(\frac{1}{x})\) for \(x \neq 0\)
(g) \(x^2 \sin(\frac{1}{x})\) for \(x \neq 0\)
(h) \(\frac{1}{x} \sin(\frac{1}{x^2})\) for \(x \neq 0\)

Accept on faith that the following familiar functions are continuous
on their domains: sin x, cos x, e^x, 2^x, loge x for x > 0, x^p for x > 0
[p any real number]. Use these facts and theorems in this section to
prove the following functions are also continuous.
(a) (1 + cos^4 x)
(b) [sin^2 x + cos^6 x]^π
(c) 2^2^x
(d) 8^x
(e) tan x for x ≠ odd multiple of (π)/(2)
(f) x sin((1)/(x)) for x ≠ 0
(g) x^2 sin((1)/(x)) for x ≠ 0
(h) (1)/(x)sin((1)/(x^2)) for x ≠ 0

Added by William J.

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