a) Using limit rules each of the following limits limx→0x+1x limx→0x+1x2 limx→0xx+1 limx→0x+1x+2 b) Using limit rules evaluate limx→2x+7−3x−2

Step 1 limx→0x+1x⇒0+10⇒10⇒∞ limx→0x+1x2⇒0+102⇒10⇒∞ limx→0xx+1⇒00+1⇒01⇒0 limx→0x+1x+2⇒0+10+2⇒12 Step 2 limx→2x+7−3x−2 ⇒limx→2(x+7−3)(x+7+3)(x−2)(x+7+3) ⇒limx→2(x+7−9)(x−2)(x+7+3) ⇒limx→2(x−2)(x−2)(x+7+3) ⇒limx→21x+7+3 ⇒12+7+3⇒19+3 ⇒13+3 ⇒16

Key to "a) Using limit rules each of the following..."

coexpennan

Answered question

2021-05-27

a) Using limit rules each of the following limits

lim_{x \to 0} \frac{x + 1}{x}

lim_{x \to 0} \frac{x + 1}{x^{2}}

lim_{x \to 0} \frac{x}{x + 1}

lim_{x \to 0} \frac{x + 1}{x + 2}

b) Using limit rules evaluate

lim_{x \to 2} \frac{\sqrt{x + 7} - 3}{x - 2}

Answer & Explanation

likvau

Skilled2021-05-28Added 75 answers

Step 1

lim_{x \to 0} \frac{x + 1}{x} \Rightarrow \frac{0 + 1}{0} \Rightarrow \frac{1}{0} \Rightarrow \infty

lim_{x \to 0} \frac{x + 1}{x^{2}} \Rightarrow \frac{0 + 1}{0^{2}} \Rightarrow \frac{1}{0} \Rightarrow \infty

lim_{x \to 0} \frac{x}{x + 1} \Rightarrow \frac{0}{0 + 1} \Rightarrow \frac{0}{1} \Rightarrow 0

lim_{x \to 0} \frac{x + 1}{x + 2} \Rightarrow \frac{0 + 1}{0 + 2} \Rightarrow \frac{1}{2}

Step 2

lim_{x \to 2} \frac{\sqrt{x + 7} - 3}{x - 2}

\Rightarrow lim_{x \to 2} \frac{(\sqrt{x + 7} - 3) (\sqrt{x + 7} + 3)}{(x - 2) (\sqrt{x + 7} + 3)}

\Rightarrow lim_{x \to 2} \frac{(x + 7 - 9)}{(x - 2) (\sqrt{x + 7} + 3)}

\Rightarrow lim_{x \to 2} \frac{(x - 2)}{(x - 2) (\sqrt{x + 7} + 3)}

\Rightarrow lim_{x \to 2} \frac{1}{\sqrt{x + 7} + 3}

\Rightarrow \frac{1}{\sqrt{2 + 7} + 3} \Rightarrow \frac{1}{\sqrt{9} + 3}

\Rightarrow \frac{1}{3 + 3}

\Rightarrow \frac{1}{6}

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