Answer to Question #275808 in Calculus for Somu

Question

a) Define Bijective function and Surjective function.

b) Find the limiting value of i) lim

𝑥→7

𝑥

2+2𝑥−63

𝑥−7

ii) lim

𝑥→∞

5𝑥−1

5𝑥+1

.

c) Test the continuity of following functions at x= -2 and x=3

𝑓(𝑥) = {

7𝑥 − 1 𝑖𝑓 𝑥 > 3

𝑥

2 − 8 𝑖𝑓 − 2 ≤ 𝑥 ≤ 3

8𝑥 + 3 𝑖𝑓 𝑥 < −2

Answer 1

a)

Bijective function is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

 Surjective function  is a function f that maps an element x to every element y; that is, for every y, there is an x such that f(x) = y.

b)

$\displaystyle\lim_{x\to 7}\frac{x^2+2x-63}{x-7}$

$x^2+2x-63=0$

$x=\frac{-2\pm \sqrt{4+252}}{2}$

$x_1=-9,x_2=7$

$x^2+2x-63=(x+9)(x-7)$

$\displaystyle\lim_{x\to 7}\frac{x^2+2x-63}{x-7}=\displaystyle\lim_{x\to 7}\frac{(x+9)(x-7)}{x-7}=\displaystyle\lim_{x\to 7}(x+9)=16$

$\displaystyle\lim_{x\to \infin}\frac{5x-1}{5x+1}=\displaystyle\lim_{x\to \infin}\frac{5-1/x}{5+1/x}=1$

c)

𝑓(𝑥) = 7𝑥 − 1 𝑖𝑓 𝑥 > 3

𝑓(𝑥) =𝑥² − 8 𝑖𝑓 − 2 ≤ 𝑥 ≤ 3

𝑓(𝑥) =8𝑥 + 3 𝑖𝑓 𝑥 < −2

for x= -2:

$\displaystyle\lim_{x\to -2^-}f(x)=\displaystyle\lim_{x\to -2^-}(8x+3)=-13$

$\displaystyle\lim_{x\to -2^+}f(x)=\displaystyle\lim_{x\to -2^-}(x^2-8)=-4$

$\displaystyle\lim_{x\to -2^-}f(x)\neq\displaystyle\lim_{x\to -2^+}f(x)$

function is not continuous

for x=3:

$\displaystyle\lim_{x\to 3^-}f(x)=\displaystyle\lim_{x\to -3^-}(x^2-8)=1$

$\displaystyle\lim_{x\to 3^+}f(x)=\displaystyle\lim_{x\to -3^+}(7x-1)=20$

$\displaystyle\lim_{x\to -3^-}f(x)\neq\displaystyle\lim_{x\to -3^+}f(x)$

function is not continuous