$$ \begin{array}{lll} ext { (j) } \lim _{x ightarrow 0} \frac{1}{\sqrt{4-x}-2} & ext { (k) } \lim _{x ightarrow 0} \frac{1}{x^{2}+3 x} & ext { (1) } \lim _{x ightarrow 0}(1-2 x)^{x} \ ext { (m) } \lim _{x ightarrow 5} \frac{2 w^{2}-50}{(w-5)(w-1)} & ext { (n) } \lim _{x ightarrow-1} \sqrt{\frac{x^{3}+x^{2}+x+1}{3 x+3}} & ext { (o) } \lim _{x ightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{1-2} ight.\end{array} $$

Question

$$ \begin{array}{lll}	ext { (j) } \lim _{x ightarrow 0} \frac{1}{\sqrt{4-x}-2} & 	ext { (k) } \lim _{x ightarrow 0} \frac{1}{x^{2}+3 x} & 	ext { (1) } \lim _{x ightarrow 0}(1-2 x)^{x} \ 	ext { (m) } \lim _{x ightarrow 5} \frac{2 w^{2}-50}{(w-5)(w-1)} & 	ext { (n) } \lim _{x ightarrow-1} \sqrt{\frac{x^{3}+x^{2}+x+1}{3 x+3}} & 	ext { (o) } \lim _{x ightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{1-2}ight.\end{array} $$

UpStudy AI · Accepted Answer

Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{1-2}\right)$$
- step1: Subtract the numbers:
  $$\lim _{x\rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{-1}\right)$$
- step2: Use the rules for multiplication and division:
  $$\lim _{x\rightarrow 0^{-}}\left(\frac{1}{x}+\frac{1}{1}\right)$$
- step3: Divide the terms:
  $$\lim _{x\rightarrow 0^{-}}\left(\frac{1}{x}+1\right)$$
- step4: Rewrite the expression:
  $$\lim _{x\rightarrow 0^{-}}\left(\frac{1}{x}\right)+\lim _{x\rightarrow 0^{-}}\left(1\right)$$
- step5: Calculate:
  $$\left(-\infty\right)+1$$
- step6: Calculate:
  $$-\infty$$
Calculate the limit $$ \lim_{x \rightarrow 0}(1-2 x)^{x} $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 0}\left(\left(1-2x\right)^{x}\right)$$
- step1: Rewrite the expression:
  $$\left(\lim _{x\rightarrow 0}\left(1-2x\right)\right)^{\lim _{x\rightarrow 0}\left(x\right)}$$
- step2: Calculate:
  $$1^{\lim _{x\rightarrow 0}\left(x\right)}$$
- step3: Calculate:
  $$1^{0}$$
- step4: Calculate:
  $$1$$
Calculate the limit $$ \lim_{w \rightarrow 5} \frac{2 w^{2}-50}{(w-5)(w-1)} $$.
Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{w\rightarrow 5}\left(\frac{2w^{2}-50}{\left(w-5\right)\left(w-1\right)}\right)$$
- step1: Reduce the fraction:
  $$\lim _{w\rightarrow 5}\left(\frac{\left(w+5\right)\times 2}{w-1}\right)$$
- step2: Rewrite the expression:
  $$\frac{\lim _{w\rightarrow 5}\left(\left(w+5\right)\times 2\right)}{\lim _{w\rightarrow 5}\left(w-1\right)}$$
- step3: Evaluate:
  $$\frac{20}{\lim _{w\rightarrow 5}\left(w-1\right)}$$
- step4: Evaluate:
  $$\frac{20}{4}$$
- step5: Reduce the numbers:
  $$\frac{5}{1}$$
- step6: Calculate:
  $$5$$
Calculate the limit $$ \lim_{x \rightarrow 0} \frac{1}{\sqrt{4-x}-2} $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 0}\left(\frac{1}{\sqrt{4-x}-2}\right)$$
- step1: Evaluate the left-hand and the right-hand limits:
  $$\begin{align}&\lim _{x\rightarrow 0^{-}}\left(\frac{1}{\sqrt{4-x}-2}\right)\&\lim _{x\rightarrow 0^{+}}\left(\frac{1}{\sqrt{4-x}-2}\right)\end{align}$$
- step2: Evaluate the left-hand limit:
  $$\begin{align}&+\infty\&-\infty\end{align}$$
- step3: Limit does not exist:
  $$\textrm{The limit does not exist}$$
Calculate the limit $$ \lim_{x \rightarrow -1} \sqrt{\frac{x^{3}+x^{2}+x+1}{3 x+3}} $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow -1}\left(\sqrt{\frac{x^{3}+x^{2}+x+1}{3x+3}}\right)$$
- step1: Simplify:
  $$\lim _{x\rightarrow -1}\left(\sqrt{\frac{x^{2}+1}{3}}\right)$$
- step2: Rewrite the expression:
  $$\sqrt{\lim _{x\rightarrow -1}\left(\frac{x^{2}+1}{3}\right)}$$
- step3: Calculate:
  $$\sqrt{\frac{2}{3}}$$
- step4: Use the properties of radicals:
  $$\frac{\sqrt{2}}{\sqrt{3}}$$
- step5: Multiply by the Conjugate:
  $$\frac{\sqrt{2}\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}$$
- step6: Multiply the numbers:
  $$\frac{\sqrt{6}}{\sqrt{3}\times \sqrt{3}}$$
- step7: Multiply the numbers:
  $$\frac{\sqrt{6}}{3}$$
Calculate the limit $$ \lim_{x \rightarrow 0} \frac{1}{x^{2}+3 x} $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 0}\left(\frac{1}{x^{2}+3x}\right)$$
- step1: Evaluate the left-hand and the right-hand limits:
  $$\begin{align}&\lim _{x\rightarrow 0^{-}}\left(\frac{1}{x^{2}+3x}\right)\&\lim _{x\rightarrow 0^{+}}\left(\frac{1}{x^{2}+3x}\right)\end{align}$$
- step2: Evaluate the left-hand limit:
  $$\begin{align}&-\infty\&+\infty\end{align}$$
- step3: Limit does not exist:
  $$\textrm{The limit does not exist}$$
Here are the results for each limit:

1. **Limit (j)**: 
   $$
   \lim_{x \rightarrow 0} \frac{1}{\sqrt{4-x}-2}
   $$
   - **Result**: The limit does not exist.

2. **Limit (k)**: 
   $$
   \lim_{x \rightarrow 0} \frac{1}{x^{2}+3x}
   $$
   - **Result**: The limit does not exist.

3. **Limit (1)**: 
   $$
   \lim_{x \rightarrow 0}(1-2x)^{x}
   $$
   - **Result**: $$1$$.

4. **Limit (m)**: 
   $$
   \lim_{w \rightarrow 5} \frac{2w^{2}-50}{(w-5)(w-1)}
   $$
   - **Result**: $$5$$.

5. **Limit (n)**: 
   $$
   \lim_{x \rightarrow -1} \sqrt{\frac{x^{3}+x^{2}+x+1}{3x+3}}
   $$
   - **Result**: $$\frac{\sqrt{6}}{3}$$.

6. **Limit (o)**: 
   $$
   \lim_{x \rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{1-2}\right)
   $$
   - **Result**: $$-\infty$$.

If you need further explanations or details on any specific limit, feel free to ask!
Here are the results for each limit: 1. **Limit (j)**: Does not exist. 2. **Limit (k)**: Does not exist. 3. **Limit (1)**: 1. 4. **Limit (m)**: 5. 5. **Limit (n)**: $$\frac{\sqrt{6}}{3}$$. 6. **Limit (o)**: $$-\infty$$.

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