Evaluate the limit: lim (x-0) [sqrt(1+x2) − 1] / x2

Question

Evaluate the limit: lim (x->0) [sqrt(1+x2) − 1] / x2

ScanSolve AI Tutor · Accepted Answer

Step 1: Check for indeterminate form by substituting x=0. lim_(x → 0) (√(1+x²) − 1)/(x²) = (√(1+0²) − 1)/(0²) = (√(1) − 1)/(0) = (1−1)/(0) = 0/0 Since this is an indeterminate form, we need to simplify the expression. Step 2: Multiply the numerator and denominator by the conjugate of the numerator. The conjugate of √(1+x²) − 1 is √(1+x²) + 1. lim_(x → 0) (√(1+x²) − 1)/(x²) × (√(1+x²) + 1)/(√(1+x²) + 1) Step 3: Simplify the numerator using the difference of squares formula, (a−b)(a+b) = a² − b². Here, a = √(1+x²) and b = 1. lim_(x → 0) ((√(1+x²))² − 1²)/(x²(√(1+x²) + 1)) = lim_(x → 0) ((1+x²) − 1)/(x²(√(1+x²) + 1)) Step 4: Further simplify the numerator. lim_(x → 0) (x²)/(x²(√(1+x²) + 1)) Step 5: Cancel out the common term x² from the numerator and denominator (since x → 0, x ≠ 0). lim_(x → 0) (1)/(√(1+x²) + 1) Step 6: Substitute x=0 into the simplified expression to evaluate the limit. (1)/(√(1+0²) + 1) = (1)/(√(1) + 1) = (1)/(1 + 1) = 1/2 The value of the limit is 1/2. The final answer is 1/2.

Evaluate the limit: lim (x->0) [sqrt(1+x2) - 1] / x2

Answer

Step-by-step solution

Check for indeterminate form by substituting $x=0$ .

Multiply the numerator and denominator by the conjugate of the numerator.

Simplify the numerator using the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$ .

Further simplify the numerator.

Cancel out the common term $x^2$ from the numerator and denominator (since $x \to 0$ , $x \neq 0$ ).

Substitute $x=0$ into the simplified expression to evaluate the limit.

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