How should I attempt this question?
Find
$$\lim_{x\to0}\frac{\cos x -1}{x^2}$$
Here are my steps. Doesn't feels correct.
$$\lim_{x\to0}\frac{\cos x -1}{x^2}=\lim_{x\to0}\frac{\cos x -1}{x}*\lim_{x\to0}\frac{1}{x}=0*\lim_{x\to0}\frac{1}{x}=0$$
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$$\lim_{x\to0}\frac{\cos x -1}{x^2}=-\lim_{x\to0}\frac{2\sin^2\frac x 2}{x^2}=-\frac 12 \lim_{x\to0}\left(\frac{\sin\frac x 2}{\frac x 2}\right)^2$$ $$=-\frac 12 \left(\lim_{x\to0}\frac{\sin\frac x 2}{\frac x 2}\right)^2=-\frac 1 2$$ For clarity, we may put $x=2y, x\to0 \implies y=\frac x 2 \to0$ $$=-\frac 12 \left(\lim_{x\to0}\frac{\sin\frac x 2}{\frac x 2}\right)^2=-\frac 12 \left(\lim_{y\to0}\frac{\sin y}{y}\right)^2=-\frac 12 $$ |
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$$\lim_{x\to0}\frac{\cos x -1}{x^2}=\lim_{x\to0}\frac{(\cos x -1)(\cos x+1)}{x^2(\cos x+1)}=\lim_{x\to0}\frac{\cos^2 x -1}{x^2(\cos x+1)}=-\lim_{x\to0}\frac{\sin^2x}{x^2}\lim_{x\to0}\frac{1}{(\cos x+1)}=-\lim_{x\to0}(\frac{\sin x}{x})^2\lim_{x\to0}\frac{1}{(\cos x+1)}=-\frac{1}{2}$$ |
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There are several mistakes in your computation, but luckily this is a very instructive counter-example! In your first step, you apply the product formula for a limit. But you can only do this when both of the factors have a limit that exists. In your case, $$ \lim_{x\to 0} \frac{1}{x} $$ does not exist (it tends to $+ \infty$), and so you can't split the limit over the product. The second mistake you make is in trying to evaluate something of the form $0 \times \infty$. There's no way to do this: what rule do you apply - zero times anything is zero, or infinity times anything is infinity? The answer is neither, and there are examples where you have expressions of the form $0 \times \infty$ that can take any real value, or $\pm \infty$. Other answers will give you an indication of some ways to evaluate this correctly, but given your original question, I thought I would offer some commentary on your solution. |
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$$\lim_{x \to a} (f(x) g(x)) = \lim_{x \to a} f(x) \lim_{x \to a} g(x)$$ only when $-\infty < \lim_{x \to a} f(x), \lim_{x \to a} g(x) < \infty$. A better way is to write the Taylor series of $\cos(x)$ as $1 - \dfrac{x^2}2 + \mathcal{O}(x^4)$. This gives us that $$\dfrac{\cos(x) - 1}{x^2} = -\dfrac12 + \mathcal{O}(x^2)$$ Hence, $$\lim_{x \to 0}\dfrac{\cos(x) - 1}{x^2} =\lim_{x \to 0}\left( -\dfrac12 + \mathcal{O}(x^2) \right) = -\dfrac12 + \lim_{x \to 0} \mathcal{O}(x^2) = -\dfrac12$$ |
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