Math / Calculus

QuestionDetermine the interval(s) on which the following function is continuous, then analyze the given limits. $h(x)=\frac{4+4 \sin x}{\cos x} ; \lim _{x \rightarrow \pi / 2^{+}} h(x) ; \lim _{x \rightarrow 5 \pi / 3} h(x)$

Studdy Solution

STEP 1

1. The function $h(x) = \frac{4 + 4 \sin x}{\cos x}$ involves trigonometric functions $\sin x$ and $\cos x$ .
2. The function is continuous wherever the denominator $\cos x$ is not zero.
3. To find the limits as $x$ approaches specific points, the behavior of both the numerator and the denominator needs to be analyzed near those points.

STEP 2

1. Determine the intervals on which $h(x)$ is continuous by identifying where $\cos x \neq 0$ .
2. Analyze the limit of $h(x)$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
3. Analyze the limit of $h(x)$ as $x$ approaches $\frac{5\pi}{3}$ .

STEP 3

Identify the points where $\cos x = 0$ since these points will cause discontinuities in the function $h(x)$ .
$\cos x = 0 \quad \text{at} \quad x = \frac{\pi}{2} + k\pi \quad \text{for} \quad k \in \mathbb{Z}$

STEP 4

Determine the intervals where $\cos x \neq 0$ .
$\text{The intervals are:} \quad (k\pi, (k+1)\pi) \quad \text{for} \quad k \in \mathbb{Z}$

STEP 5

Analyze the limit of $h(x)$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
$\lim_{x \to \frac{\pi}{2}^{+}} h(x) = \lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \sin x}{\cos x}$

STEP 6

Examine the behavior of $\sin x$ and $\cos x$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
$\sin \left( \frac{\pi}{2}^{+} \right) = 1 \quad \text{and} \quad \cos \left( \frac{\pi}{2}^{+} \right) \to 0^{-}$

STEP 7

Calculate the limit considering the behavior of the trigonometric functions.
$\lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \sin x}{\cos x} = \lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \cdot 1}{0^{-}} = \frac{8}{0^{-}} = -\infty$
Thus,
$\lim_{x \to \frac{\pi}{2}^{+}} h(x) = -\infty$

STEP 8

Analyze the limit of $h(x)$ as $x$ approaches $\frac{5\pi}{3}$ .
$\lim_{x \to \frac{5\pi}{3}} h(x) = \lim_{x \to \frac{5\pi}{3}} \frac{4 + 4 \sin x}{\cos x}$

STEP 9

Determine the values of $\sin x$ and $\cos x$ at $x = \frac{5\pi}{3}$ .
$\sin \left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2} \quad \text{and} \quad \cos \left( \frac{5\pi}{3} \right) = \frac{1}{2}$

STEP 10

Calculate the value of $h(x)$ at $x = \frac{5\pi}{3}$ .
$h \left( \frac{5\pi}{3} \right) = \frac{4 + 4 \left( -\frac{\sqrt{3}}{2} \right)}{\frac{1}{2}} = \frac{4 - 2\sqrt{3}}{\frac{1}{2}} = 8 - 4\sqrt{3}$
Thus,
$\lim_{x \to \frac{5\pi}{3}} h(x) = 8 - 4\sqrt{3}$
Solution: The function $h(x)$ is continuous on the intervals $(k\pi, (k+1)\pi)$ for $k \in \mathbb{Z}$ , except at the points $x = \frac{\pi}{2} + k\pi$ . The limits are:
$\lim_{x \to \frac{\pi}{2}^{+}} h(x) = -\infty$
$\lim_{x \to \frac{5\pi}{3}} h(x) = 8 - 4\sqrt{3}$

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QuestionDetermine the interval(s) on which the following function is continuous, then analyze the given limits. $h(x)=\frac{4+4 \sin x}{\cos x} ; \lim _{x \rightarrow \pi / 2^{+}} h(x) ; \lim _{x \rightarrow 5 \pi / 3} h(x)$

Studdy Solution

STEP 1

STEP 2

1. Determine the intervals on which $h(x)$ is continuous by identifying where $\cos x \neq 0$ .
2. Analyze the limit of $h(x)$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
3. Analyze the limit of $h(x)$ as $x$ approaches $\frac{5\pi}{3}$ .

STEP 3

Identify the points where $\cos x = 0$ since these points will cause discontinuities in the function $h(x)$ .
$\cos x = 0 \quad \text{at} \quad x = \frac{\pi}{2} + k\pi \quad \text{for} \quad k \in \mathbb{Z}$

STEP 4

Determine the intervals where $\cos x \neq 0$ .
$\text{The intervals are:} \quad (k\pi, (k+1)\pi) \quad \text{for} \quad k \in \mathbb{Z}$

STEP 5

Analyze the limit of $h(x)$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
$\lim_{x \to \frac{\pi}{2}^{+}} h(x) = \lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \sin x}{\cos x}$

STEP 6

Examine the behavior of $\sin x$ and $\cos x$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
$\sin \left( \frac{\pi}{2}^{+} \right) = 1 \quad \text{and} \quad \cos \left( \frac{\pi}{2}^{+} \right) \to 0^{-}$

STEP 7

Calculate the limit considering the behavior of the trigonometric functions.
$\lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \sin x}{\cos x} = \lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \cdot 1}{0^{-}} = \frac{8}{0^{-}} = -\infty$
Thus,
$\lim_{x \to \frac{\pi}{2}^{+}} h(x) = -\infty$

STEP 8

Analyze the limit of $h(x)$ as $x$ approaches $\frac{5\pi}{3}$ .
$\lim_{x \to \frac{5\pi}{3}} h(x) = \lim_{x \to \frac{5\pi}{3}} \frac{4 + 4 \sin x}{\cos x}$

STEP 9

Determine the values of $\sin x$ and $\cos x$ at $x = \frac{5\pi}{3}$ .
$\sin \left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2} \quad \text{and} \quad \cos \left( \frac{5\pi}{3} \right) = \frac{1}{2}$

STEP 10

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Studdy Solution

STEP 1

STEP 2

1. Determine the intervals on which h(x) h(x) h(x) is continuous by identifying where cos⁡x≠0\cos x \neq 0cosx=0.2. Analyze the limit of h(x) h(x) h(x) as x x x approaches π2+ \frac{\pi}{2}^{+} 2π​+.3. Analyze the limit of h(x) h(x) h(x) as x x x approaches 5π3 \frac{5\pi}{3} 35π​.

STEP 3

Identify the points where cos⁡x=0\cos x = 0cosx=0 since these points will cause discontinuities in the function h(x) h(x) h(x).cos⁡x=0atx=π2+kπfork∈Z\cos x = 0 \quad \text{at} \quad x = \frac{\pi}{2} + k\pi \quad \text{for} \quad k \in \mathbb{Z}cosx=0atx=2π​+kπfork∈Z

STEP 4

Determine the intervals where cos⁡x≠0\cos x \neq 0cosx=0.The intervals are:(kπ,(k+1)π)fork∈Z\text{The intervals are:} \quad (k\pi, (k+1)\pi) \quad \text{for} \quad k \in \mathbb{Z}The intervals are:(kπ,(k+1)π)fork∈Z

STEP 5

Analyze the limit of h(x) h(x) h(x) as x x x approaches π2+ \frac{\pi}{2}^{+} 2π​+.lim⁡x→π2+h(x)=lim⁡x→π2+4+4sin⁡xcos⁡x\lim_{x \to \frac{\pi}{2}^{+}} h(x) = \lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \sin x}{\cos x}limx→2π​+​h(x)=limx→2π​+​cosx4+4sinx​

STEP 6

Examine the behavior of sin⁡x\sin xsinx and cos⁡x\cos xcosx as x x x approaches π2+\frac{\pi}{2}^{+}2π​+.sin⁡(π2+)=1andcos⁡(π2+)→0−\sin \left( \frac{\pi}{2}^{+} \right) = 1 \quad \text{and} \quad \cos \left( \frac{\pi}{2}^{+} \right) \to 0^{-}sin(2π​+)=1andcos(2π​+)→0−

STEP 7

STEP 8

Analyze the limit of h(x) h(x) h(x) as x x x approaches 5π3 \frac{5\pi}{3} 35π​.lim⁡x→5π3h(x)=lim⁡x→5π34+4sin⁡xcos⁡x\lim_{x \to \frac{5\pi}{3}} h(x) = \lim_{x \to \frac{5\pi}{3}} \frac{4 + 4 \sin x}{\cos x}limx→35π​​h(x)=limx→35π​​cosx4+4sinx​

STEP 9

STEP 10

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1. Determine the intervals on which $h(x)$ is continuous by identifying where $\cos x \neq 0$ .
2. Analyze the limit of $h(x)$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
3. Analyze the limit of $h(x)$ as $x$ approaches $\frac{5\pi}{3}$ .

Identify the points where $\cos x = 0$ since these points will cause discontinuities in the function $h(x)$ .
$\cos x = 0 \quad \text{at} \quad x = \frac{\pi}{2} + k\pi \quad \text{for} \quad k \in \mathbb{Z}$

Determine the intervals where $\cos x \neq 0$ .
$\text{The intervals are:} \quad (k\pi, (k+1)\pi) \quad \text{for} \quad k \in \mathbb{Z}$

Analyze the limit of $h(x)$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
$\lim_{x \to \frac{\pi}{2}^{+}} h(x) = \lim_{x \to \frac{\pi}{2}^{+}} \frac{4 + 4 \sin x}{\cos x}$

Examine the behavior of $\sin x$ and $\cos x$ as $x$ approaches $\frac{\pi}{2}^{+}$ .
$\sin \left( \frac{\pi}{2}^{+} \right) = 1 \quad \text{and} \quad \cos \left( \frac{\pi}{2}^{+} \right) \to 0^{-}$

Analyze the limit of $h(x)$ as $x$ approaches $\frac{5\pi}{3}$ .
$\lim_{x \to \frac{5\pi}{3}} h(x) = \lim_{x \to \frac{5\pi}{3}} \frac{4 + 4 \sin x}{\cos x}$