M_B
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How can we define the limit of a constant function?
7 votes

Let $f(x)=c$, where $c$ is some constant number, and suppose that the domain of $f$ is the real numbers. Then we can take the limit as $x$ approaches some value, say for example $$\lim_{x\rightarrow0}...

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Good books on complex numbers
4 votes

Books on complex analysis definitely use the topics that you mentioned, but usually assume that the reader is already familiar with some algebra and geometry of complex numbers. The book Visual ...

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Why $\sin\left( \frac 1 x \right) $ oscillates infinitely many times as $x \to 0$
3 votes

You have shown that if there is a period, then it must be infinitesimally small, which implies that the function under consideration must be constant. However, you don't consider the case where there ...

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What is the purpose of a function being surjective?
3 votes

The definition of a function is not just the rule $f$. Instead, a function is defined by a domain $A$ and a codomain $B$ together with a rule $f$ that takes every $x\in A$ and returns a unique element ...

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How to calculate limit as x approaches infinity of a^x/b^x?
3 votes

Hint: review properties of exponents, especially that the quotient of two numbers each raised to the the $x$ power is equal to the quotient of the two numbers itself raised to the $x$ power.

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Is the intersection of these given sequences the empty set?
2 votes

Hint: Assume some $x>0$ is in the intersection. Can you find a contradiction to this assumption? That is, can you find one of the intervals in the intersection not containing $x$? Responding to ...

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Clarification on proof for the product form of a sum on a completely multiplicative function
2 votes

Assume to the contrary that there is a prime $p$ with $f(p) \geq 1$. Then $f(p^{k})=f(p)^{k}$, $\forall k \in \mathbb{N}$, a contradiction, since then there are infinitely many values of $f$ greater ...

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sum (difference) of polynomials to the power n
1 votes

By the binomial theorem $$ (f_1+f_2)^n -(f_1-f_2)^n = \sum_{k=0}^{n}\binom{n}{k}f_1^{n-k}f_2^k -\sum_{k=0}^{n}\binom{n}{k}f_1^{n-k}(-f_2)^k. $$ Notice that since we have $(-f_2)^k$ the second sum is ...

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How to tell a dimension of a subspace
Accepted answer
1 votes

The dimension theorem for vector spaces states that all bases of a vector space have the same number of elements. So if you can determine the cardinality of one basis, you have determined the ...

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Describe Mathematically and graphically the region represented by $\ -\pi \lt arg (z) \lt \pi $
1 votes

Recall that the argument of a complex number is the angle that $z\in\mathbb{C}$ forms in standard position. Considering the set $\{z\in\mathbb{C}:-\pi< \arg (z)<\pi\}$, note the radius is not ...

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Unconnected sets.
1 votes

Let $B_{r}(z)$ be the open ball centered at $z\in\mathbb{C}$ with radius $r$. Consider the two open balls $B_{1}(-1)$ and $B_{1}(1)$. The first is the set of points with $|z+1|<1$ and the second is ...

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What does "min" mean?
1 votes

Note that the minimum is not always well-defined. For example, given the interval $(0,1)$, there is no smallest element in the interval, since we may find numbers in the interval arbitrarily close to ...

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How to prove that if a function $f$ from a finite set $X$ to $X$ is surjective, then $f$ is injective.
0 votes

Use the fact that the domain and range have the same number of elements. Since every element in the range is hit by the function when we assume surjectivity and every element in the domain is sent to ...

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How to prove the following.
Accepted answer
0 votes

I will show you how to get started on a proof by induction. Base case: for $n=0$, we have $a_0=1$ by definition, and $2^{0+1}-1=1$. Induction step: Suppose $a_k=2^{k+1}-1$ for some $k>0$. Then ...

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Closure of an open set
0 votes

I will prove that the only topology that satisfies the given property is the discrete topology. Assume that $(X,T)$ is a topological space such that if $A$ and $B$ are open sets with $A$ a proper ...

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Trigonometric Equation : $\sin 96^\circ \sin 12^\circ \sin x = \sin 18^\circ \sin 42^\circ \sin (12^\circ -x)$
0 votes

Hint: have you tried using the product-to-sum identities? That is, $$ \sin A\sin B =\frac{1}{2}(\cos(A-B)-\cos(A+B)) $$ is one such formula. Note that there are several formulas of this type that can ...

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