Determine: 1.1 lim (x->1) (x/ln x - 1/ln x) 1.2 lim (x->inf) x sin (1/x) 2 If f(x) = (2x)/(1-sin 2x) Find the value(s) of x that will make the function f(x) to be discontinuous for x ∈ [0°; 90°]

June 22, 2026Mathematics

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June 22, 2026

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Step 1: Combine the fractions. $\lim_{x \to 1} \left( \frac{x}{\ln x} - \frac{1}{\ln x} \right) = \lim_{x \to 1} \frac{x-1}{\ln x}$ Step 2: Evaluate the limit. Substituting $x=1$ gives $\frac{1-1}{\ln 1} = \frac{0}{0}$ , which is an indeterminate form. Apply L'Hôpital's Rule. $\lim_{x \to 1} \frac{\frac{d}{dx}(x-1)}{\frac{d}{dx}(\ln x)} = \lim_{x \to 1} \frac{1}{\frac{1}{x}}$ Step 3: Simplify and substitute $x=1$ . $\lim_{x \to 1} x = 1$ The final answer is $\boxed{1}$ .

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Combine the fractions. _x 1 ( (x)/( x) - (1)/( x) ) = _x 1 (x-1)/( x) Step 2: Evaluate the limit.

Asked on June 22, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on June 22, 2026

Step 1: Combine the fractions. _x 1 ( (x)/( x) - (1)/( x) ) = _x 1 (x-1)/( x) Step 2: Evaluate the limit. Substituting x=1 gives (1-1)/( 1) = (0)/(0), which is an indeterminate form. Apply L'Hôpital's Rule. _x 1 (d)/(dx)(x-1)(d)/(dx)( x) = _x 1 (1)/(1)x Step 3: Simplify and substitute x=1. _x 1 x = 1 The final answer is 1. Send me the next one 📸