2. Compute each of the following limits. \( \begin{array}{lll}\text { a) } \lim _{x \rightarrow \infty} \frac{1}{x} & \text { d) } \lim _{x \rightarrow-\infty}\left(\frac{-5}{2 x^{3}}-7+\frac{8}{x}\right) & \text { g) } \lim _{x \rightarrow \infty} \frac{-5 x^{3}-2 x+4}{x^{2}} \\ \text { b) } \lim _{x \rightarrow-\infty} \frac{1}{x} & \text { e) } \lim _{x \rightarrow \infty}\left(-2 x^{3}+1-\frac{5}{x}+\frac{12}{x^{4}}\right) & \text { h) } \lim _{x \rightarrow \infty} \frac{-5 x^{3}-2 x+4}{x^{3}} \\ \text { c) } \lim _{x \rightarrow \infty} \frac{-5}{2 x^{3}} & \text { f) } \lim _{x \rightarrow-\infty} \frac{3 x-2}{x} & \text { i) } \lim _{x \rightarrow \infty} \frac{-5 x^{3}-2 x+4}{x^{4}}\end{array} \)

Let's compute each of the given limits step by step. --- ### **a)** \[ \lim_{x \rightarrow \infty} \frac{1}{x} \] **Solution:** As \( x \) approaches infinity, \( \frac{1}{x} \) approaches 0. **Answer:** \[ 0 \] --- ### **b)** \[ \lim_{x \rightarrow -\infty} \frac{1}{x} \] **Solution:** As \( x \) approaches negative infinity, \( \frac{1}{x} \) also approaches 0 (from the negative side, but the limit is still 0). **Answer:** \[ 0 \] --- ### **c)** \[ \lim_{x \rightarrow \infty} \frac{-5}{2x^{3}} \] **Solution:** As \( x \) approaches infinity, the denominator \( 2x^3 \) grows without bound, making the entire fraction approach 0. **Answer:** \[ 0 \] --- ### **d)** \[ \lim_{x \rightarrow -\infty} \left(\frac{-5}{2x^{3}} - 7 + \frac{8}{x}\right) \] **Solution:** - \( \frac{-5}{2x^3} \) approaches 0 as \( x \) approaches negative infinity. - \( -7 \) is a constant. - \( \frac{8}{x} \) also approaches 0 as \( x \) approaches negative infinity. Combining these: \[ 0 - 7 + 0 = -7 \] **Answer:** \[ -7 \] --- ### **e)** \[ \lim_{x \rightarrow \infty} \left(-2x^{3} + 1 - \frac{5}{x} + \frac{12}{x^{4}}\right) \] **Solution:** The term \( -2x^3 \) dominates as \( x \) approaches infinity. The other terms become negligible in comparison. **Answer:** \[ -\infty \] --- ### **f)** \[ \lim_{x \rightarrow -\infty} \frac{3x - 2}{x} \] **Solution:** Simplify the expression: \[ \frac{3x - 2}{x} = 3 - \frac{2}{x} \] As \( x \) approaches negative infinity, \( \frac{2}{x} \) approaches 0. \[ 3 - 0 = 3 \] **Answer:** \[ 3 \] --- ### **g)** \[ \lim_{x \rightarrow \infty} \frac{-5x^{3} - 2x + 4}{x^{2}} \] **Solution:** Divide each term by \( x^2 \): \[ \frac{-5x^3}{x^2} + \frac{-2x}{x^2} + \frac{4}{x^2} = -5x - \frac{2}{x} + \frac{4}{x^2} \] As \( x \) approaches infinity: - \( -5x \) approaches \( -\infty \). - \( \frac{-2}{x} \) approaches 0. - \( \frac{4}{x^2} \) approaches 0. **Answer:** \[ -\infty \] --- ### **h)** \[ \lim_{x \rightarrow \infty} \frac{-5x^{3} - 2x + 4}{x^{3}} \] **Solution:** Divide each term by \( x^3 \): \[ \frac{-5x^3}{x^3} + \frac{-2x}{x^3} + \frac{4}{x^3} = -5 - \frac{2}{x^2} + \frac{4}{x^3} \] As \( x \) approaches infinity: - \( -5 \) remains constant. - \( \frac{-2}{x^2} \) approaches 0. - \( \frac{4}{x^3} \) approaches 0. \[ -5 + 0 + 0 = -5 \] **Answer:** \[ -5 \] --- ### **i)** \[ \lim_{x \rightarrow \infty} \frac{-5x^{3} - 2x + 4}{x^{4}} \] **Solution:** Divide each term by \( x^4 \): \[ \frac{-5x^3}{x^4} + \frac{-2x}{x^4} + \frac{4}{x^4} = \frac{-5}{x} + \frac{-2}{x^3} + \frac{4}{x^4} \] As \( x \) approaches infinity, each term approaches 0. **Answer:** \[ 0 \] --- ## **Summary of Results** | **Part** | **Limit** | **Result** | |----------|-----------------------------------------|---------------| | a) | \( \lim_{x \rightarrow \infty} \frac{1}{x} \) | \( 0 \) | | b) | \( \lim_{x \rightarrow -\infty} \frac{1}{x} \) | \( 0 \) | | c) | \( \lim_{x \rightarrow \infty} \frac{-5}{2x^{3}} \) | \( 0 \) | | d) | \( \lim_{x \rightarrow -\infty} \left(\frac{-5}{2x^{3}} - 7 + \frac{8}{x}\right) \) | \( -7 \) | | e) | \( \lim_{x \rightarrow \infty} \left(-2x^{3} + 1 - \frac{5}{x} + \frac{12}{x^{4}}\right) \) | \( -\infty \) | | f) | \( \lim_{x \rightarrow -\infty} \frac{3x - 2}{x} \) | \( 3 \) | | g) | \( \lim_{x \rightarrow \infty} \frac{-5x^{3} - 2x + 4}{x^{2}} \) | \( -\infty \) | | h) | \( \lim_{x \rightarrow \infty} \frac{-5x^{3} - 2x + 4}{x^{3}} \) | \( -5 \) | | i) | \( \lim_{x \rightarrow \infty} \frac{-5x^{3} - 2x + 4}{x^{4}} \) | \( 0 \) |