Limits and Continuity

This is a classic 'math joke' or visual puzzle. If you treat 'n' as a variable in the expression 'sin x / n' and 'cancel' the 'n' in the numerator and denominator, you are left with 'six', which is 6. Mathematically, the limit is actually undefined/infinite as n approaches 0.

Using the Taylor series expansion, tan(x) = x + x^3/3 + ... near x = 0. Substituting: tan(x) - x = x^3/3. Dividing by x^3 gives 1/3. The limit exists and equals 1/3 as x approaches 0.

$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ = $\lim_{x \to \infty}\frac{1- \dfrac{\sin x}{x}}{1+\dfrac{\cos x}{x}}$

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The ROC of addition or subtraction of two functions $x_1(n) \ and \ x_2(n)$ is $R_1 \cap R_2$. We have been given ROC of addition of two function and has been asked ROC of subtraction of two function. It will be same.

$lim_{\theta \rightarrow 0} \frac{sin(\theta / 2)}{\theta} = lim_{\theta \rightarrow 0} \frac{sin(\theta / 2)}{2(\theta / 2)} = \frac{1}{2} lim_{\theta \rightarrow 0} \frac{sin(\theta / 2)}{(\theta / 2)} = \frac{1}{2} = 0.5$

Simpson's rule for 1/x from 0.5 to 1.5 with three points (h=0.5): x=0.5, 1.0, 1.5. Integral = (h/3) * (f(0.5) + 4f(1.0) + f(1.5)) = (0.5/3) * (2 + 4 + 0.666) = 0.1666 * 6.666 = 1.111. Exact value = ln(1.5) - ln(0.5) = ln(3) = 1.0986. Difference = 1.111 - 1.0986 = 0.0124.

Using the limit property lim(x->0) sin(ax)/x = a, we substitute a = 2/3. Thus, the limit is 2/3.

Limit x->0 of (x^3 + x^2) / (2x^3 - 7x^2). Divide numerator and denominator by x^2: (x + 1) / (2x - 7). As x->0, this becomes 1 / -7 = -1/7.