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Home Question Bank Math Calculus Archive 2025 January 18

Calculus Questions from Jan 18,2025

Browse the Calculus Q&A Archive for Jan 18,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. \( \lim _{x \rightarrow 1} \frac{\ln x}{20 x-x^{2}-19}=\lim _{x \rightarrow 1}(\square) \) Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. \( \lim _{x \rightarrow-3} \frac{x^{2}+7 x+12}{-18-3 x+x^{2}}=\lim _{x \rightarrow-3} \) Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. \( \lim _{x \rightarrow-3} \frac{x^{2}+7 x+12}{-18-3 x+x^{2}} \) Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. \( \lim _{x \rightarrow-3} \frac{x^{2}+7 x+12}{-18-3 x+x^{2}}=\lim _{x \rightarrow-3} \) Evaluate the limit. \( \lim _{x \rightarrow-3} \frac{x^{2}+7 x+12}{-18-3 x+x^{2}}=\square \) (Type an exact answer.) In terms of limits, what does it mean for \( f \) to grow faster than \( g \) as \( x \rightarrow \infty \) ? Choose the correct answer below. A. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=0 \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\infty \) B. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=f(x) \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=g(x) \) O. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=\infty \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0 \) O. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=g(x) \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=f(x) \) Explain why the form \( 1^{\infty} \) is indeterminate and cannot be evaluated by substitution. Explain how the competing functions behave. Choose the correct answer. A. If \( \lim _{x \rightarrow a} f(x)=1 \) and \( \lim _{x \rightarrow a} g(x)=\infty \), then \( f(x)^{g(x)} \rightarrow 0 \) as \( x \rightarrow a \), so the limit does not exist. Therefore, substitution cannot be used. B. If \( \lim _{x \rightarrow a} f(x)=1 \) and \( \lim _{x \rightarrow a} g(x)=\infty \), then \( f(x)^{g(x)} \rightarrow 0^{0} \) as \( x \rightarrow a \), which is meaningless. Therefore, substitution cannot be used. C. If \( \lim _{x \rightarrow a} f(x)=1 \) and \( \lim _{x \rightarrow a} g(x)=\infty \), then \( f(x)^{g(x)} \rightarrow 1^{\infty} \) as \( x \rightarrow a \), which is meaningless. Therefore, substitution cannot be used. OD. If \( \lim _{x \rightarrow a} f(x)=1 \) and \( \lim _{x \rightarrow a} g(x)=\infty \), then \( f(x)^{g(x)} \rightarrow \infty \) as \( x \rightarrow a \), so the limit does not exist. Therefore, substitution cannot be used. Explain how to convert a limit of the form \( 0 \cdot \infty \) to a limit of the form \( 0 / 0 \) or \( \infty / \infty 0 \). Suppose that \( \lim _{x \rightarrow a} f(x) g(x) \) has the indeterminate form \( 0 \cdot \infty \), where \( \lim _{x \rightarrow a} f(x)=0 \) and \( \lim _{x \rightarrow a} g(x)=\infty \). Rewrite the limit as \( \lim _{x \rightarrow a} \) (in the form \( \infty / \infty \) ) and apply \( \quad \) (in the form \( 0 / 0 \) ) or as \( \lim _{x \rightarrow a} \) (in quotient, if possible. Explain the steps used to apply l'Hôpital's Rule to a limit of the form \( \frac{0}{0} \). Choose the correct answer. A. Take the limit of the quotient of the derivatives of the numerator and denominator. B. Take the limit of the quotient of the numerator and denominator. C. Rewrite the quotient as a product, then take the limit of the derivative of the product. D. Take the limit of the derivative obtained using the Quotient Rule. Explain with examples what is meant by the indeterminate form \( \frac{0}{0} \). A. If \( \lim _{x \rightarrow a} f(x)=\infty \) and \( \lim _{x \rightarrow a} g(x)=\infty \), then \( \lim _{x \rightarrow a} \frac{f(x)}{g(x)} \) is of indeterminate form \( \frac{0}{0} \). For example, set \( f(x)=x^{-1} \) and \( g(x)=\frac{1}{\sin ^{2} x} \) and \( a=0 \). B. If \( \lim _{x \rightarrow a} f(x)=0 \) and \( \lim _{x \rightarrow a} g(x)=0 \), then \( \lim _{x \rightarrow a} \frac{f(x)}{g(x)} \) is of indeterminate form \( \frac{0}{0} \). For example, set \( f(x)=x \) and \( g(x)=\sin x \) and \( a=0 \). C. If \( \lim _{x \rightarrow a} f(x)=0 \) and \( \lim _{x \rightarrow a} g(x)=0 \), then \( \lim _{x \rightarrow a} f(x)^{g(x)} \) is of indeterminate form \( \frac{0}{0} \). For example, set \( f(x)=x \) and \( g(x)=\sin x \) and \( a=0 \). D. If \( \lim _{x \rightarrow a} f(x)=0 \) and \( \lim _{x \rightarrow a} g(x)=\infty \), then \( \lim _{x \rightarrow a} f(x) g(x) \) is of indeterminate form \( \frac{0}{0} \). For example, set \( f(x)=x \) and \( g(x)=\frac{1}{\sin ^{2} x} \) and \( a=0 \). ind the equation of tangent and normal to the curve \( x=t^{2}-t, y=3 t+ \) at the point \( (2,10) \). \( y = [ x + ( x + \sin ^ { 2 } x ) ^ { 3 } ] ^ { 4 } \)
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