Investigating whether a function is bounded

The graph of the function is symmetric about the $y$-axis, so we only need to think about the lower bound when $x\geq 0$. And $y$ is increasing for $x\geq 0$. So the minimum occurs when $x=0$, so the greatest lower bound is $-\dfrac{3}{7}$.


We have

$$y=\frac{x^{2}-3}{x^{2}+7}=1-\frac{10}{x^{2}+7}\ge 1-\frac{10}7$$

Tags:

Calculus

Real Analysis

Related

If $A$ and $B$ are groups then prove that $A\times B\cong B\times A$ What is a practical use for this metric? Find $x \in \mathbb{R}$ such that $(x^2+p^2)/xp < -2$ Prove $R(a \times b) = Ra \times Rb$ given $R \in \mathcal{SO}(3)$ and $a,b \in \mathbb{R}^3$. Prove the Rational Limit Theorem. When are the eigenvalues of a matrix containing all squared elements irrational/rational? Functional equation problem: $ f \left( y ^ 2 - f ( x ) \right) = y f ( x ) ^ 2 + f \left( x ^ 2 y + y \right) $ How to change the order of integration when limit is a function? Simplifying $i \left[ \ln(x+i)-\ln(x-i)-\ln(1+ix)+\ln(1-ix) \right]$ The identity $(u\times v)\cdot(x\times y)=\begin{vmatrix}u\cdot x&v\cdot x\\ u\cdot y&v\cdot y\\\end{vmatrix}$ Is power set functor determined by its image on objects? Find the maximum number of triangles that can have equal area

Recent Posts