We read your photo as:

Evaluate the limit: lim (x-> -2) (x+2) / (sqrt(x+6) - 2)

Mathematics

ScanSolve AI Answer

Answer

4

Step-by-step solution

5 steps

  1. Substitute x=−2x = -2 into the expression to check for an indeterminate form.

    −2+2−2+6−2=04−2=02−2=00\frac{-2+2}{\sqrt{-2+6}-2} = \frac{0}{\sqrt{4}-2} = \frac{0}{2-2} = \frac{0}{0}

    Since this is an indeterminate form, we need to simplify the expression.

  2. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is x+6+2\sqrt{x+6}+2.

    lim⁡x→−2x+2x+6−2×x+6+2x+6+2\lim_{x \to -2} \frac{x+2}{\sqrt{x+6}-2} \times \frac{\sqrt{x+6}+2}{\sqrt{x+6}+2}

  3. Multiply the terms.

    Use the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2-b^2 for the denominator.

    lim⁡x→−2(x+2)(x+6+2)(x+6)2−22\lim_{x \to -2} \frac{(x+2)(\sqrt{x+6}+2)}{(\sqrt{x+6})^2 - 2^2} lim⁡x→−2(x+2)(x+6+2)x+6−4\lim_{x \to -2} \frac{(x+2)(\sqrt{x+6}+2)}{x+6-4} lim⁡x→−2(x+2)(x+6+2)x+2\lim_{x \to -2} \frac{(x+2)(\sqrt{x+6}+2)}{x+2}

  4. Cancel out the common factor (x+2)(x+2) from the numerator and denominator.

    lim⁡x→−2(x+6+2)\lim_{x \to -2} (\sqrt{x+6}+2)

  5. Substitute x=−2x = -2 into the simplified expression.

    −2+6+2=4+2=2+2=4\sqrt{-2+6}+2 = \sqrt{4}+2 = 2+2 = 4

    The limit is 4.

    The final answer is 4.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QEvaluate the limit: lim (x->0) [sqrt(2x+3) - sqrt(3)] / xQEvaluate the limit: lim (x->0) [sqrt(1+x) - sqrt(1-x)] / xQEvaluate the limit: lim (2x − 1) * sqrt(x+4) as x approaches 0.QEvaluate the limit: lim x->0 (sqrt(x2+9)−3)/x2 and lim x->1 (e^(x−1)−1)/(x−1)2QEvaluate the limit: lim (t->0) x / (3 - sqrt(x+9))
Browse by topic:Calculus
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp