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f(x)=cos 2theta

f(x)=cos⁡2θ f(x)=\cos 2 \theta

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Full solution

Q. f(x)=cos⁡2θ f(x)=\cos 2 \theta
  1. Simplify the expression: We start by simplifying the expression inside the limit: \newline1x+4−14\frac{1}{x+4}-\frac{1}{4} over xx\newlineFirst, find a common denominator for the fractions in the numerator:\newline1x+4−14\frac{1}{x+4} - \frac{1}{4} = 4−(x+4)4(x+4)\frac{4 - (x+4)}{4(x+4)}\newline= −x4(x+4)\frac{-x}{4(x+4)}
  2. Find common denominator: Next, simplify the entire expression by dividing the numerator by xx:−x4(x+4)÷x=−x4x(x+4)\frac{-x}{4(x+4)}\div x = \frac{-x}{4x(x+4)}=−14(x+4)= \frac{-1}{4(x+4)}
  3. Divide numerator by xx: Now, evaluate the limit as xx approaches 00:lim⁡x→0(−14(x+4))=−14(0+4)=−116\lim_{x \to 0}\left(\frac{-1}{4(x+4)}\right) = \frac{-1}{4(0+4)} = \frac{-1}{16}

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