Knight Gough
01/09/2024 · Primary School
\( \lim _{x \rightarrow +\infty } (\ln (x+\sin (x))-3\ln x \)
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Step-by-step Solution
The limit provided is not supported for direct calculation. Let's simplify the expression first before calculating the limit.
Given expression: \( \lim _{x \rightarrow +\infty } (\ln (x+\sin (x))-3\ln x) \)
We can simplify this expression by using the properties of logarithms.
\( \ln (x+\sin (x)) = \ln x + \ln (1+\frac{\sin (x)}{x}) \)
As \( x \) approaches infinity, \( \frac{\sin (x)}{x} \) approaches 0. Therefore, \( \ln (1+\frac{\sin (x)}{x}) \) approaches 0.
So, the expression simplifies to: \( \lim _{x \rightarrow +\infty } (\ln x + \ln (1+\frac{\sin (x)}{x}) - 3\ln x) \)
\( = \lim _{x \rightarrow +\infty } (\ln (1+\frac{\sin (x)}{x}) - 2\ln x) \)
Now, we can calculate the limit of this simplified expression.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
\(\lim _{x\rightarrow +\infty}\left(\ln{\left(1+\frac{\sin\left(x\right)}{x}\right)}-2\ln{\left(x\right)}\right)\)
- step1: Simplify:
\(\lim _{x\rightarrow +\infty}\left(\ln{\left(1+x^{-1}\sin\left(x\right)\right)}-2\ln{\left(x\right)}\right)\)
- step2: Rewrite the expression:
\(\lim _{x\rightarrow +\infty}\left(\ln{\left(1+x^{-1}\sin\left(x\right)\right)}\right)+\lim _{x\rightarrow +\infty}\left(-2\ln{\left(x\right)}\right)\)
- step3: Calculate:
\(0+\left(-\infty\right)\)
- step4: Calculate:
\(-\infty\)
The limit of the simplified expression \( \lim _{x \rightarrow +\infty } (\ln (1+\frac{\sin (x)}{x}) - 2\ln x) \) is \( -\infty \).
Quick Answer
The limit is \( -\infty \).
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