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Chapter 2: Problem 60

Let \(\begin{aligned} f(x) &=\cos x, & & x \geq 0 \\ &=x+k, & & x<0 \end{aligned}\) Find the value of constant \(k\), given that \(\lim _{x \rightarrow 0} f(x)\) exists. \\{Ans. \(\left.k=1\right\\}\)

Short Answer

Expert verified
The value of the constant \(k\) is 1, ensuring the left and right limits of the function as x approaches 0 are equal and allowing the limit to exist.

Step by step solution

01

Find the left limit of the function as x approaches 0

Since the function is defined as \(f(x) = x + k\) for \(x < 0\), the left limit is: \[\lim_{x\to0^-}(x+k)\]
02

Find the right limit of the function as x approaches 0

For \(x \geq 0\), the function is defined as \(f(x) = \cos x\). Therefore, the right limit is: \[\lim_{x\to0^+}\cos x\]
03

Set the left limit equal to the right limit and solve for \(k\)

For the limit to exist, both left and right limits should be equal, i.e., \[\lim_{x\to0^-}(x+k) = \lim_{x\to0^+}\cos x\] Since \(x\) approaches 0 in the left limit, we substitute 0 in the equation: \[0+k = \lim_{x\to0^+}\cos x\] We know that \(\lim_{x\to0^+}\cos x\) is also equal to \(\cos 0\): \[k = \cos 0\] We know that \(\cos 0 = 1\): \[k = 1\] Therefore, the value of the constant \(k\) is 1.

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