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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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Evaluate the following limit. Use IHôpital's Rule when it is convenient and applicable. \[ \lim _{x \rightarrow \frac{\pi}{}_{-}^{-}} \frac{\tan x}{\left(\frac{20}{2 x-\pi}\right)} \] How should the given limit be evaluated? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Use l'Hôpital's Rule to rewrite the limit as lim \( \square \) \[ x \rightarrow \frac{\pi^{-}}{2} \] B. Multiply the expression by a unit fraction to obtain lim \( \square \) ). \[ x \rightarrow \frac{\pi^{-}}{2} \] C. Use a trigonometric identity to rewrite the limit as \( \lim \) \( \square \) ). \[ x_{x \rightarrow \frac{\pi^{-}}{2}} \] D. Use direct substitution. Evaluate the limit. \[ \lim _{u \rightarrow \frac{\pi}{4}} \frac{8 \tan u-8 \cot u}{u-\frac{\pi}{4}}=\square \text { (Type an exact answer.) } \] Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. \( \lim _{u \rightarrow \frac{\pi}{4}} \frac{8 \tan u-8 \cot \mathrm{u}}{u-\frac{\pi}{4}}=\lim _{u \rightarrow \frac{\pi}{4}} \) Evaluate the limit. \( \lim _{x \rightarrow 0} \frac{4 \sin 9 x}{7 x}=\square \) (Type an exact answer.) Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. \( \lim _{x \rightarrow 0} \frac{4 \sin 9 x}{7 x}=\lim _{x \rightarrow 0} \) Evaluate the limit. \( \lim _{x \rightarrow e} \frac{2 \ln x-2}{x-e}=\square \) (Type an exact answer.) Use l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form. \( \lim _{x \rightarrow e} \frac{2 \ln x-2}{x-e}=\lim _{x \rightarrow e}(\square) \) Attempt all numbers in the paper 1) If \( y=\frac{\cos x}{x^{2}} \), prove that; \( x^{2} \frac{d^{2} y}{d x^{2}}+4 x \frac{d y}{d x}+\left(2+x^{2}\right) y=0 \) 2) Given the parametric equations \( x=3+4 \cos \alpha, y=5-8 \sin \alpha \). Find \( \frac{d^{2} y}{d x^{2}} \). Evaluate the following limit. Use lHôpital's Rule when it is convenient and applicable. \( \lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x} \), for some constant a \( \lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x}=\square \) (Type an exact answer.) Evaluate the limit. \( \lim _{x \rightarrow 1} \frac{\ln x}{20 x-x^{2}-19}=\square \) IType an exact answer.)
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