1. Evaluate \( \displaystyle{\lim_{x \to \infty} \frac{\sin x}{x}} \) View Solution
2. Find the limit \(\displaystyle{\lim_{x \to -\infty} \left( 3 + \frac{200}{x} + \frac{\sin x }{x^2}\right)}\) View Solution
3. Find the limit \( \displaystyle{\lim_{x \to \infty} \frac{x^2-5x+7}{4x^2+3x+1}} \) View Solution
4. Find the limit \( \displaystyle{\lim_{x \to \infty} \frac{3x^4-2x+5}{4x^3-5x+1}}\) View Solution
5. Find the limit \( \displaystyle{\lim_{x \to -\infty} \frac{1-2x+4x^3}{2+x^2-3x^3}}\) View Solution
6. Find the limit \(\displaystyle{\lim_{x \to \infty} \frac{3x^2-2x+5}{4x^2 + 2x+1}}\) View Solution
7. Find the limit \(\displaystyle{\lim_{x \to \infty} \frac{\sqrt{9x^4+5}}{5x^2 -3x+1}}\) View Solution
8. Evaluate \( \displaystyle{\lim_{x \to \infty} \frac{1+e^x}{1-3e^x}} \) View Solution
9. Find the limit \( \displaystyle{\lim_{x \to -\infty} \frac{2x}{\sqrt{x^2-1}}}\) View Solution
10. Find the horizontal and vertical asymptotes of the curve:   \(\displaystyle{y=\frac{5+x^{3}}{x-x^{3}}}\) View Solution
11. Find the horizontal and vertical asymptotes of the curve:   \(\displaystyle{f(x)=\frac{\sqrt{2 x^{2}+1}}{5x-3}}\) View Solution
12. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions: View Solution \( \displaystyle{\lim_{x \to 3} f(x)=-\infty, \quad \lim_{x \to \infty} f(x)=2, \quad \lim_{x \to -\infty} f(x)=0, \quad \lim_{x \to 0^{+}} f(x)= \infty, \quad \lim_{x \to 0^{-}} f(x)=-\infty} \)
13. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions: View Solution \( \displaystyle{ f(0)=3, \quad \lim_{x \to 0^-} f(x)=4, \quad \lim_{x \to 0^+} f(x)= 2, } \)
    \( \displaystyle{ \lim_{x \to -\infty} f(x)= -\infty, \quad \lim_{x \to 4^-} f(x)=-\infty, \quad \lim_{x \to 4^+} f(x)= \infty, \quad \lim_{x \to \infty} f(x)=3 } \)
14. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions: View Solution \( \displaystyle{ f(2)=0, \quad \lim_{x \to 2^-} f(x)=\infty, \quad \lim_{x \to 2^+} f(x)=-\infty, \quad \lim_{x \to \infty} f(x)= -1, \quad \lim_{x \to -\infty} f(x)= 0} \)
15. Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions: View Solution \( \displaystyle{f(0) =0, \quad \lim_{x \to -1 ^-} f(x)=\infty, \quad \lim_{x \to -1 ^+} f(x)=-\infty, \quad \lim_{x \to 1 ^-} f(x)=-\infty, } \)
    \( \displaystyle{ \lim_{x \to 1 ^+} f(x)=\infty, \quad \lim_{x \to \infty} f(x)= 1, \quad \lim_{x \to -\infty} f(x)= -2} \)
16. Evaluate the following limits: View Solution (a) \( \displaystyle{ \lim_{x\to \infty} \arctan(x^2) } \)   (b) \( \displaystyle{ \lim_{x\to -\infty} \arctan(e^x) } \)