Final answer to the problem
Step-by-step Solution
How should I solve this problem?
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- Solve using L'Hôpital's rule
- Solve without using l'Hôpital
- Solve using limit properties
- Solve using direct substitution
- Solve the limit using factorization
- Solve the limit using rationalization
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
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The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete
Simplify the numerators
Combine and simplify all terms in the same fraction with common denominator $\left(x-1\right)\ln\left(x\right)$
If we directly evaluate the limit $\lim_{x\to 1}\left(\frac{x\ln\left(x\right)-x+1}{\left(x-1\right)\ln\left(x\right)}\right)$ as $x$ tends to $1$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
After deriving both the numerator and denominator, the limit results in
If we directly evaluate the limit $\lim_{x\to 1}\left(\frac{\ln\left(x\right)}{\ln\left(x\right)+1+\frac{-1}{x}}\right)$ as $x$ tends to $1$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
After deriving both the numerator and denominator, the limit results in
Evaluate the limit $\lim_{x\to1}\left(\frac{x}{1+x}\right)$ by replacing all occurrences of $x$ by $1$