# Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate. lim_{trightarrow0}frac{5t^2}{cos t-1}

Question
Limits and continuity
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$

2021-01-17
Consider the limit:
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$
If we try to evaluate the limit by direct substitution we get 0/0.
So, evaluate using l’Hôpital’s Rule
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}=5\lim_{t\rightarrow0}(\frac{\frac{d}{dt}(t^2)}{\frac{d}{dt}(\cos t-1)})$$
$$=5\lim_{t\rightarrow0}(\frac{2t}{-\sin t})$$
We het 0/0 form, so apply again L'Hospital Rule
$$5\lim_{t\rightarrow0}(\frac{\frac{d}{dt}(2t)}{\frac{d}{dt}(-\sin t)})=5\lim_{t\rightarrow0}(\frac{2}{-\cos t})$$
$$=5(\frac{2}{-\cos0})$$
$$=5(-\frac{2}{-1})$$
$$=-10$$

### Relevant Questions

Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}$$

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Evaluate the limits using algebra and|or limit properties as needed
Use the limit definition of the derivative to calculate the derivatives of the following function:
(a) $$f(x)=4x^2+3x+1$$
(b) $$f(x)=\frac{2}{x^2}$$
Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.
a) $$\lim \frac{2n+1}{5n+4}=\frac{2}{5}$$
b) $$\lim \frac{2n^3}{n^3+3}=0$$
c) $$\lim \frac{\sin (n^2)}{\sqrt[3]{n}}$$
Suppose the functions f(x) and g(x) are defined for all x and that $$\lim_{x\rightarrow0}f(x)=\frac{1}{2}$$ and $$\lim_{x\rightarrow0}g(x)=\sqrt2$$. Find the limits as $$x\rightarrow0$$ of the following functions. $$f(x)\frac{\cos x}{x-1}$$
Let $$a_n\rightarrow0$$, and use the Algebraic Limit Theorem to compute each of the following limits (assuming the fractions are always defined):
$$\lim_{n\rightarrow\infty}\frac{1+2a_n}{1+3a_n-4(a_n)^2}$$
Find the following limits or state that they do not exist. $$\lim_{w\rightarrow1}\frac{a\cdot1}{(w^2-2)}-\frac{1}{(w-1)b}$$
Use Taylor series to evaluate the following limits.
$$\lim_{x\rightarrow0}\frac{\sec x-\cos x-x^2}{x^4} \ (Hint: \text{The Maclaurin series for sec x is }1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+...)$$

When a gas is taken from a to c along the curved path in the figure (Figure 1) , the work done by the gas is W = -40 J and the heat added to the gas is Q = -140 J . Along path abc, the work done by the gas is W = -50 J . (That is, 50 J of work is done on the gas.)
I keep on missing Part D. The answer for part D is not -150,150,-155,108,105( was close but it said not quite check calculations)
Part A
What is Q for path abc?
Express your answer to two significant figures and include the appropriate units.
Part B
f Pc=1/2Pb, what is W for path cda?
Express your answer to two significant figures and include the appropriate units.
Part C
What is Q for path cda?
Express your answer to two significant figures and include the appropriate units.
Part D
What is Ua?Uc?
Express your answer to two significant figures and include the appropriate units.
Part E
If Ud?Uc=42J, what is Q for path da?
Express your answer to two significant figures and include the appropriate units.
...