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Chapter 1: Problem 34

\(\lim _{x \rightarrow-\infty} x+\sqrt{x^{2}+x^{2} \sin (1 / x)}\) is equal to (A) 0 (B) 2 (C) \(-2\) (D) none of these

Short Answer

Expert verified
Answer: The limit of the expression as x approaches negative infinity is \(-2\).

Step by step solution

01

Identify the dominating term

As x approaches negative infinity, we can observe that the dominating term in the expression is \(x^2\). We can then focus on finding the limit with respect to this term.
02

Factor out \(x^2\) from the expression

Let's factor out \(x^2\) from the expression inside the square root: \(\lim _{x \rightarrow-\infty} x+\sqrt{x^{2}(1+\sin (1 / x ))}\)
03

Simplify the expression further

Now, we can rewrite the above expression into a simpler form: \(\lim _{x \rightarrow-\infty} x+\sqrt{x^{2}}\sqrt{(1+\sin (1 / x ))}\)
04

Analyze the behavior of \(\sin(1/x)\) as x approaches negative infinity

As x approaches negative infinity, \(1/x\) approaches 0, which means the argument of the sine function oscillates between close to -1 and close to 1. Thus, we can write the inequality: -1 鈮� \(\sin (1 / x)\) 鈮� 1 which gives us the inequality for the expression inside the square root: 0 鈮� \(1+\sin (1 / x)\) 鈮� 2
05

Apply the square root to the inequality

Since the square root function is monotonic, we can take the square root of each term of the inequality: \(\sqrt{0}\) 鈮� \(\sqrt{1+\sin (1 / x)}\) 鈮� \(\sqrt{2}\)
06

Find the limit of the expression

Now we can take the limit of each term of the inequality: \(\lim _{x \rightarrow-\infty} x+\sqrt{x^{2}}\sqrt{0}\) 鈮� \(\lim _{x \rightarrow-\infty} x+\sqrt{x^{2}}\sqrt{(1+\sin (1 / x ))}\) 鈮� \(\lim _{x \rightarrow-\infty} x+\sqrt{x^{2}}\sqrt{2}\)
07

Simplify and evaluate the limit

As x approaches negative infinity, the limit of each term yields the following: \(-2\) 鈮� \(-\infty + \sqrt{x^2}\sqrt{(1+\sin (1 / x ))}\) 鈮� \(-\infty + \sqrt{2x^2}\) Since the middle term is trapped between -2 and -鈭� as x approaches negative infinity, we can conclude that the limit is equal to -2. Hence, the correct answer is (C) \(-2\).

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Most popular questions from this chapter

The true statement(s) is / are (A) If \(\mathrm{f}(\mathrm{x})<\mathrm{g}(\mathrm{x})\) for all $\mathrm{x} \neq \mathrm{a}\(, then \)\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})<\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})$. (B) If \(\lim _{x \rightarrow c} \mathrm{f}(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c\), then \(\lim f(x) \cdot g(x)=0\) (C) If \(\lim _{x \rightarrow c} \mathrm{f}(\mathrm{x})=\mathrm{L}\), then $\lim _{\mathrm{x} \rightarrow \mathrm{c}}|\mathrm{f}(\mathrm{x})|=|\mathrm{L}|$ and conversely if \(\lim |\mathrm{f}(\mathrm{x})|=|\mathrm{L}|\) then $\lim _{x \rightarrow \infty} \mathrm{f}(\mathrm{x})=\mathrm{L}$. (D) If \(\mathrm{f}(\mathrm{x})=\mathrm{g}(\mathrm{x})\) for all real number other then \(\mathrm{x}=0\) and \(\lim _{x \rightarrow 0} f(x)=L\), then $\lim _{x \rightarrow 0} g(x)=L$

$\lim _{x \rightarrow 0} \lim _{x \rightarrow 0} \frac{2(\tan x-\sin x)-x^{3}}{x^{5}}$ is equal to (A) \(1 / 4\) (B) \(1 / 2\) (C) \(1 / 3\) (D) None of these

Column - I (A) If $\lim _{x \rightarrow \infty}\left(\sqrt{\left(x^{2}-x-1\right)}-a x-b\right)=0\(, where \)a>0$, then there exists atleast one a and \(b\) for which point (a, \(2 b\) ) lies on the line (B) If $\lim _{x \rightarrow 0} \frac{\left(1+a^{3}\right)+8 e^{1 / x}}{1+\left(1-b^{3}\right) e^{1 / x}}=2$, then there exists atleast one \(a\) and \(b\) for which point \(\left(a, b^{3}\right)\) lies on the line (C) If $\lim _{\mathrm{x} \rightarrow \infty}\left(\sqrt{\left(\mathrm{x}^{4}-\mathrm{x}^{2}+1\right)}-\mathrm{ax}^{2}-\mathrm{b}\right)=0$, then there exists atleast one a and \(\mathrm{b}\) for which point \((\mathrm{a},-2 \mathrm{~b})\) lies on the line (D) If \(\lim _{x \rightarrow-a} \frac{x^{7}+a^{7}}{x+a}=7\), where \(a<0\), then there exists atleast one a for which point \((a, 2)\) lies on the line. Column - II (P) \(\mathrm{y}=-3\) (Q) \(3 x-2 y-5=0\) (R) \(15 x-2 y-13=0\) (S) \(\mathrm{y}=2\)

Column - I (A) $\lim _{n \rightarrow \infty} \cos ^{2}\left(\pi\left(\sqrt[3]{n^{3}+n^{2}+2 n}\right)\right)\( where \)n$ is an integer, equals. (B) $\lim _{n \rightarrow \infty} \mathrm{n} \sin \left(2 \pi \sqrt{1+\mathrm{n}^{2}}\right)(\mathrm{n} \in \mathrm{N})$ equals. (C) $\lim _{n \rightarrow \infty}(-1)^{n} \sin \left(\pi \sqrt{n^{2}+0.5 n+1}\right)\left(\sin \frac{(n+1) \pi}{2 n}\right)\( is (where \)\left.n \in N\right)$. (D) If \(\lim _{x \rightarrow \infty}\left(\frac{x+a}{x-a}\right)^{x}=e\) where 'a' is some real constant then the value of 'a' is equal to. Column - II (P) \(\frac{1}{\sqrt{2}}\) (Q) \(\frac{1}{4}\) (R) \(\pi\) (S) non existent

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