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Hi I’m not sure how to solve this can you help me? I’m in high school Calculus 1, and this is a homework. Thanks!

Hi I’m not sure how to solve this can you help me? I’m in high school Calculus 1, and-example-1
User Cesans
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1 Answer

6 votes

Given:


f(x)=\begin{cases}x^3+4x^2+1,\text{ }ifx\le-4 \\ \sqrt[]{x+5\text{ }},\text{ if }-4a) x=-4.<p>The condition for the function f(x) continuity at x=a is </p>[tex]\lim _(x\to a^+)f(x)=\lim _(x\to a^-)f(x)=f(a)

Set x=-4, Consider the limit.


\lim _(x\to-4^+)f(x)=\lim _(x\to-4^+)\sqrt[]{x+5}


\lim _(x\to-4^+)f(x)=\sqrt[]{-4+5}


\lim _(x\to-4^+)f(x)=1

Consider the limit.


\lim _(x\to-4^-)f(x)=\lim _(x\to-4^-)(x^3+4x^2+1)


\lim _(x\to-4^-)f(x)=(-4)^3+4(-4)^2+1


\lim _(x\to-4^-)f(x)=-64+64+1


\lim _(x\to-4^-)f(x)=1

Hence we get


\lim _(x\to-4^-)f(x)=\lim _(x\to-4^+)f(x)=1

x= -4 satisfies the continuity condition for the given function.

Hence x= -4 is the continuity point.

b) x=1.

Set x=1, Consider the limit.


\lim _(x\to1^+)f(x)=\lim _(x\to1^+)\sqrt[]{x+5}


\lim _(x\to1^+)f(x)=\sqrt[]{1+5}


\lim _(x\to1^+)f(x)=\sqrt[]{6}

Consider the limit.


\lim _(x\to1^-)f(x)=\lim _(x\to1^-)\sqrt[]{x+5}


\lim _(x\to1^-)f(x)=\sqrt[]{1+5}


\lim _(x\to1^-)f(x)=\sqrt[]{6}

Hence we get


\lim _(x\to1^-)f(x)=\lim _(x\to1^+)f(x)=\sqrt[]{6}

x=1 satisfies the continuity of the given function f(x).

Hence x= 1 is the continuity point.

c) x=3pi/4

Consider the limit.


\lim _{x\to(3\pi)/(4)^+}f(x)=\lim _{x\to(3\pi)/(4)^+}\sin x


\lim _{x\to(3\pi)/(4)^+}f(x)=\sin ((3\pi)/(4))


\lim _{x\to(3\pi)/(4)^+}f(x)=\sin (\pi-(\pi)/(4))


\lim _{x\to(3\pi)/(4)^+}f(x)=\sin ((\pi)/(4))


\lim _{x\to(3\pi)/(4)^+}f(x)=\frac{1}{\sqrt[]{2}}


\lim _{x\to(3\pi)/(4)^+}f(x)=0.707

Consider the limit.


\lim _{x\to(3\pi)/(4)^-}f(x)=\lim _{x\to(3\pi)/(4)^-}\sqrt[]{x+5}


\lim _{x\to(3\pi)/(4)^-}f(x)=\sqrt[]{(3\pi)/(4)+5}
\text{ Use }\pi=180.


\lim _{x\to(3\pi)/(4)^-}f(x)=\sqrt[]{(3*180)/(4)+5}


\lim _{x\to(3\pi)/(4)^-}f(x)=\sqrt[]{140}


\lim _{x\to(3\pi)/(4)^-}f(x)=11.83

Hence we get


0.707\\e11.83


\lim _{x\to(3\pi)/(4)^+}f(x)\\e\lim _{x\to(3\pi)/(4)^-}f(x)

The point x=3pi/4 does not satisfy the continuity of the function.

Hence x=3pi/4 is the discontinuity point of the given function.

The graph of the function is

Hi I’m not sure how to solve this can you help me? I’m in high school Calculus 1, and-example-1
User Dave Carlile
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