suppose\n\$\\lim_{x\\to a}h(x)=1, \\lim_{x\\to a}f(x)=1, \\lim_{x\\to a}g(x)=0\$.\nfind following limits…

suppose\n\$\\lim_{x\\to a}h(x)=1, \\lim_{x\\to a}f(x)=1, \\lim_{x\\to a}g(x)=0\$.\nfind following limits if they exist. enter dne if the limit does not exist.\n1. \$\\lim_{x\\to a}h(x)+f(x)\$\n2. \$\\lim_{x\\to a}h(x)-f(x)\$\n3. \$\\lim_{x\\to a}h(x)\\cdot g(x)\$\n4. \$\\lim_{x\\to a}\\frac{h(x)}{f(x)}\$\n5. \$\\lim_{x\\to a}\\frac{h(x)}{g(x)}\$\n6. \$\\lim_{x\\to a}\\frac{g(x)}{h(x)}\$\n7. \$\\lim_{x\\to a}(f(x))^{2}\$\n8. \$\\lim_{x\\to a}\\frac{1}{f(x)}\$\n9. \$\\lim_{x\\to a}\\frac{1}{f(x)-g(x)}\$

Answer

Explanation:

Step1: Use limit - sum rule

$\lim_{x\rightarrow a}(h(x)+f(x))=\lim_{x\rightarrow a}h(x)+\lim_{x\rightarrow a}f(x)=1 + 1=2$

Step2: Use limit - difference rule

$\lim_{x\rightarrow a}(h(x)-f(x))=\lim_{x\rightarrow a}h(x)-\lim_{x\rightarrow a}f(x)=1 - 1=0$

Step3: Use limit - product rule

$\lim_{x\rightarrow a}(h(x)\cdot g(x))=\lim_{x\rightarrow a}h(x)\cdot\lim_{x\rightarrow a}g(x)=1\times0 = 0$

Step4: Use limit - quotient rule

$\lim_{x\rightarrow a}\frac{h(x)}{f(x)}=\frac{\lim_{x\rightarrow a}h(x)}{\lim_{x\rightarrow a}f(x)}=\frac{1}{1}=1$

Step5: Since $\lim_{x\rightarrow a}g(x)=0$ and $\lim_{x\rightarrow a}h(x)=1$

$\lim_{x\rightarrow a}\frac{h(x)}{g(x)}=\text{DNE}$

Step6: Use limit - quotient rule

$\lim_{x\rightarrow a}\frac{g(x)}{h(x)}=\frac{\lim_{x\rightarrow a}g(x)}{\lim_{x\rightarrow a}h(x)}=\frac{0}{1}=0$

Step7: Use limit - power rule

$\lim_{x\rightarrow a}(f(x))^{2}=(\lim_{x\rightarrow a}f(x))^{2}=1^{2}=1$

Step8: Use limit - reciprocal rule

$\lim_{x\rightarrow a}\frac{1}{f(x)}=\frac{1}{\lim_{x\rightarrow a}f(x)}=\frac{1}{1}=1$

Step9: First, find $\lim_{x\rightarrow a}(f(x)-g(x))$

$\lim_{x\rightarrow a}(f(x)-g(x))=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}g(x)=1-0 = 1$, then $\lim_{x\rightarrow a}\frac{1}{f(x)-g(x)}=\frac{1}{1}=1$

suppose\n\\(\\lim_{x\\to a}h(x)=1, \\lim_{x\\to a}f(x)=1, \\lim_{x\\to a}g(x)=0\\).\nfind following limits…