guess the value of the limit (if it exists) by evaluating the function at the given numbers. (it is…
guess the value of the limit (if it exists) by evaluating the function at the given numbers. (it is suggested that you report answers accurate to at least six decimal places.) let $f(x)=\frac{cos(10x)-cos(12x)}{x^{2}}$. we want to find the limit $lim_{x \to 0}\frac{cos(10x)-cos(12x)}{x^{2}}$. start by calculating the values of the function for the inputs listed in this table. based on the values in this table, it appears $lim_{x \to 0}\frac{cos(10x)-cos(12x)}{x^{2}}=$. question help: message instructor submit question jump to answer
Answer
Explanation:
Step1: Evaluate for $x = 0.2$
Substitute $x = 0.2$ into $f(x)=\frac{\cos(10x)-\cos(12x)}{x^{2}}$. $f(0.2)=\frac{\cos(10\times0.2)-\cos(12\times0.2)}{(0.2)^{2}}=\frac{\cos(2)-\cos(2.4)}{0.04}\approx\frac{- 0.416147 - (-0.737394)}{0.04}=\frac{-0.416147 + 0.737394}{0.04}=\frac{0.321247}{0.04}=8.031175$
Step2: Evaluate for $x = 0.1$
Substitute $x = 0.1$ into $f(x)$. $f(0.1)=\frac{\cos(10\times0.1)-\cos(12\times0.1)}{(0.1)^{2}}=\frac{\cos(1)-\cos(1.2)}{0.01}\approx\frac{0.540302-0.362358}{0.01}=\frac{0.177944}{0.01}=17.794400$
Step3: Evaluate for $x = 0.05$
Substitute $x = 0.05$ into $f(x)$. $f(0.05)=\frac{\cos(10\times0.05)-\cos(12\times0.05)}{(0.05)^{2}}=\frac{\cos(0.5)-\cos(0.6)}{0.0025}\approx\frac{0.877583 - 0.825336}{0.0025}=\frac{0.052247}{0.0025}=20.898800$
Step4: Evaluate for $x = 0.01$
Substitute $x = 0.01$ into $f(x)$. $f(0.01)=\frac{\cos(10\times0.01)-\cos(12\times0.01)}{(0.01)^{2}}=\frac{\cos(0.1)-\cos(0.12)}{0.0001}\approx\frac{0.995004 - 0.992806}{0.0001}=\frac{0.002198}{0.0001}=21.980000$
Step5: Evaluate for $x = 0.001$
Substitute $x = 0.001$ into $f(x)$. $f(0.001)=\frac{\cos(10\times0.001)-\cos(12\times0.001)}{(0.001)^{2}}=\frac{\cos(0.01)-\cos(0.012)}{1\times10^{- 6}}\approx\frac{0.999950 - 0.999928}{1\times10^{-6}}=\frac{0.000022}{1\times10^{-6}}=22.000000$
Step6: Evaluate for $x = 0.0001$
Substitute $x = 0.0001$ into $f(x)$. $f(0.0001)=\frac{\cos(10\times0.0001)-\cos(12\times0.0001)}{(0.0001)^{2}}=\frac{\cos(0.001)-\cos(0.0012)}{1\times10^{-8}}\approx\frac{0.9999995 - 0.99999928}{1\times10^{-8}}=\frac{0.00000022}{1\times10^{-8}}=22.000000$
Step7: Evaluate for $x = 0.00001$
Substitute $x = 0.00001$ into $f(x)$. $f(0.00001)=\frac{\cos(10\times0.00001)-\cos(12\times0.00001)}{(0.00001)^{2}}=\frac{\cos(0.0001)-\cos(0.00012)}{1\times10^{-10}}\approx\frac{0.999999995 - 0.9999999928}{1\times10^{-10}}=\frac{0.0000000022}{1\times10^{-10}}=22.000000$
Answer:
| $x$ | $f(x)$ |
|---|---|
| $0.2$ | $8.031175$ |
| $0.1$ | $17.794400$ |
| $0.05$ | $20.898800$ |
| $0.01$ | $21.980000$ |
| $0.001$ | $22.000000$ |
| $0.0001$ | $22.000000$ |
| $0.00001$ | $22.000000$ |
| $\lim_{x\rightarrow0}\frac{\cos(10x)-\cos(12x)}{x^{2}} = 22.000000$ |