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United Arab Emirates
\( 10 x-1=15-6 x \) \( \frac{3 x}{2}+5=\frac{5 x}{2}-1 \) b) \( 1 \frac{1}{2} z-2=3 \frac{1}{4} z-9 \) c) d) \( 9 x-8=11 x-10 \)
Algebra Jan 20, 2025
Find \( \lim _{x \rightarrow 0^{+}}\left(e^{-1 / x} \sin (1 / x)-(x+2)^{3}\right) \) (if it exists) and give a careful argument showing that your answer is correct. The notation lim the uniqueness of limits. Prove that limits, if they exist, are indeed unique. That is, the suppose that \( f \) is a real valued function of a real variable, \( a \) is an accumulation point of the domain of \( f \), and \( \ell, m \in \mathbb{R} \). Prove that if \( f(x) \rightarrow \ell \) as \( x \rightarrow a \) and \( f(x) \rightarrow m \) as \( x \rightarrow a \), then \( l=m \). (Explain carefully why it was important that we require \( a \) to be an accumulation point of the domain of \( f \).) Let \( f(x)=\frac{\sin \pi x}{x+1} \) for all \( x \neq-1 \). The following information is known about a function \( g \) defined for all real numbers \( x \neq 1 \) : (i) \( g=\frac{p}{q} \) where \( p(x)=a x^{2}+b x+c \) and \( q(x)=d x+e \) for some constants \( a, b, c, d, e \); (ii) the only \( x \)-intercept of the curve \( y=g(x) \) occurs at the origin; (iii) \( g(x) \geq 0 \) on the interval \( [0,1) \) and is negative elsewhere on its domain; (iv) \( g \) has a vertical asymptote at \( x=1 \); and (v) \( g(1 / 2)=3 \). Either find lim \( g(x) f(x) \) or else show that this limit does not exist. Hints. Write an explicit formula for \( g \) by determining the constants \( a \ldots e \). Use (ii) to find \( c \); use (ii) and (iii) to find \( a \); use (iv) to find a relationship between \( d \) and \( e \); then use (v) to obtain an explicit form for \( g \). Finally look at \( f(x) g(x) \); replace sin \( \pi x \) by sin( \( (x(x-1)+\pi) \) and use the formula for the sine of the sum of two numbers.
Calculus Jan 20, 2025
c) \( 9 x-8=11 x-10 \)
Algebra Jan 20, 2025
a) \( 1 \frac{1}{2} z-2=3 \frac{1}{4} z-9 \)
Algebra Jan 20, 2025
\( \frac{3 x}{2}+5=\frac{5 x}{2}-1 \)
Algebra Jan 20, 2025
\( 10 x-1=15-6 x \)
Algebra Jan 20, 2025
\( \frac{3 x}{2}+5=\frac{5 x}{2}-1 \)
Algebra Jan 20, 2025
\( 10 x-1=15-6 x \)
Algebra Jan 20, 2025
b) \( 1 \frac{1}{2} z-2=3 \frac{1}{4} z-9 \)
Algebra Jan 20, 2025
a) \( 10 x-1=15-6 x \) \( \frac{3 x}{2}+5=\frac{5 x}{2}-1 \)
Algebra Jan 20, 2025
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