Question

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.lim⁡x→01−cos⁡(x)cos⁡2(x)−3cos⁡(x)+2{\displaystyle\lim_{x	o0}\frac{1-\cos\left(xight)}{\$$cos^2$$\left(xight)-3\cos\left(xight)+2}}

Accepted Answer

Step 1: Recognize that the limit involves a trigonometric expression. The numerator is $$1 - \cos(x)$$, which can be approximated using the Taylor series expansion for $$\cos(x)$$ around $$x = 0$$: $$\cos(x) \approx 1 - \frac{x^2$}{2}. $$Therefore, $$1 - \cos(x) \approx \frac{x^2$}{2}$$ for small $$x. $$Step 2: Factor the denominator $$\cos^2$(x) - 3\cos(x) + 2. $$Notice that this is a quadratic in terms of $$\cos(x). $$Let $$y = \cos(x)$$, then the expression becomes $$y^2 - 3y + 2$. Factor this quadratic to get (y - 1)(y - 2$$)$$. Therefore, $$\$$cos^2(x) - 3$$\cos(x) + 2 = (\cos(x) - 1)(\cos(x) - 2)$$. Step 3: Substitute the factored form of the denominator back into the limit expression: $$\lim_{x 	o 0} \frac{1 - \cos(x)}{(\cos(x) - 1)(\cos(x) - 2)}$$. Notice that $$\cos(x) - 1$$ is a common factor in both the numerator and the denominator. Step 4: Simplify the expression by canceling the common factor $$\cos(x) - 1$$ from the numerator and the denominator. This gives $$\lim_{x 	o 0} \frac{1}{\cos(x) - 2}$$. Step 5: Evaluate the simplified limit as x$ approaches 0. Since $$\cos(0) = 1$$, substitute x = 0$ into the simplified expression to find the limit.

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\lim_{x \to 0} \frac{1 - \cos (x)}{\cos^{2} (x) - 3 \cos (x) + 2}$

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주요 개념

Limits

Trigonometric Functions

Indeterminate Forms

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.limx→01−cos(x)cos2(x)−3cos(x)+2{\(\displaystyle\]\lim\)_{x\(\to\)0}\(\frac{1-\cos\left(x\right)}{\cos^2\left(x\right)-3\cos\left(x\right)+2}\)}