Given below are two statements:
**Statement I:**
x→0limx5tan−1x+loge(x1+x−1−x)−2x=52
**Statement II:**
x→1lim(x1−x2)=e21
In the light of the above statements, choose the correct answer from the options given below:
Get the complete, step-by-step math solution for: "Given below are two statements: **Statement I:** _{x 0} {tan^{-1}x + _e {√(1 + x) - √(1 - x)}{x} - 2x}{x^5} = (2)/(5) **Statement II:** _{x 1} {2}{x^{...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Evaluate Statement I using Taylor series expansions
To evaluate the limit in Statement I, we will use Taylor series expansions for each term around x=0. We need to expand up to the x5 term because the denominator is x5. The expansions for tan−1x, 1+x, and loge(1+y) are shown.
Step 2: Simplify the logarithmic term in Statement I
First, we simplify the argument of the logarithm. We substitute the Taylor expansions for 1+x and 1−x and simplify the expression. Then, we substitute this simplified expression into the Taylor expansion for loge(1+y) to get the expansion for the logarithmic term.
Step 3: Substitute expansions into Statement I and evaluate the limit
Now, we substitute the Taylor series expansions for tan−1x and the logarithmic term back into the original limit expression. We also include the −2x term. After combining like terms, we can see that the numerator does not have an x5 term with a non-zero coefficient that would result in 2/5. The lowest power of x in the numerator after simplification is x, which means the limit will be undefined or zero, not 2/5. Therefore, Statement I is false.
Step 4: Evaluate Statement II using properties of limits
For Statement II, we need to evaluate the limit as x→1. We can directly substitute x=1 into the expression since the function is continuous at x=1. Substituting x=1 gives 11−1=10=1. Therefore, the limit is 2/1=2. Since the statement claims the limit is 1/e2, Statement II is false.