Limits Test

1.  Determine \( \lim_{x\to -2} \left(3x+x^2 \right) \).

2. Determine \( \lim_{x \to -2} f(x), \: \textrm{where} \: f(x)= \begin{cases}  3, & \text{if} \; x = -2 \\ -3, & \text{if} \; x \neq -2    \end{cases} \)

3. Determine an expression, in simplified form, for the slope of the secant PQ with \( P(1, 2) \) and \( Q(1 + h, f(1 + h)) \), where \( f(x) = 2x^2 \).

4. Consider the following graph of a function \( f(x) \). Determine \( \lim_{x \to 1} f(x) \).

5. Consider the difference quotient: \( \frac{2(3+h)^4 – 162}{h} \). Which statements are true?

6. The function \( y=f(x) \):

7. The function \( y = f(x) \) has a removable discontinuity at:

8. The value at \(\; f(7) \) is:

9. The value of \( \lim_{x \to 1^+} f(x) \) is

10. Which of the following is not a true statement about limits?

11. A man drops a penny from the top of a 500 m tall building. After t seconds, the penny   has  fallen  a  distance  of  s  metres,  where \( s(t)=500 – 5t^2, \; 0\leq t \leq 10 \). The velocity at \( t=5 \) is

12. Determine the slope of the tangent to \( f(x)=\frac{2}{\sqrt{x \, + \, 5} \) at the point where \( x=5 \).

13. Evaluate the limit: $$ \lim_{x \to -5^+} \frac{5x}{x + 5} $$

14. Evaluate the limit, if it exists: $$ \lim_{x \to 1} \frac{x^2 – 3x + 2}{x \, – 1}   $$

15. Evaluate the limit: $$ \lim_{x \to 125} \frac{125 – x}{x^{\frac{1}{3}} – 5} $$

16. Consider the function \( f(x) = \begin{cases}  \frac{x^2 – 9x + 20}{x – 5}, & \text{if} \; x \neq 5 \\ -1, & \text{if} \; x = 5    \end{cases}  \).

Determine \( \lim_{x \to 5} f(x)  \).

17. The difference quotient finds the

18. This function:

19. This function:

20. Complete the sentence by filling in the blank.

21. A soccer ball is kicked into the air from a platform. The height of the ball, in metres, \( t \)seconds after it is kicked is modeled by \( h(t) = -4.9t^2 + 13.5t + 1.2 \). Find the instantaneous rate of change 2 seconds.

22. Determine the coordinates of the point on the curve \(  y = \sqrt{x – 1} \) , where the tangent is perpendicular to the line \(  4x + y + 1 = 0  \).

23. Given \( f(x)= \begin{cases}  \;  ax^2 + bx + c, & \text{if} \; x \lt 2 \\ (2 – x)^3 + 10x, & \text{if} \; x \ge 2    \end{cases} \)

\( f(x) \) is continuous at \( x = 2 \), has a y-intercept of 4, and at \( x = 1 \) the tangent to the curve is horizontal. Use these characteristics to find a, b, and c.

24. $$ \textrm{Evaluate} \; \lim_{x \to -3^5} \frac{(x^{\frac{3}{5}} + 27)( x^{\frac{1}{5}} +3)^2}{x^{\frac{3}{5}} + 9x^{\frac{2}{5}} + 27x^{\frac{1}{5}} + 27 }   $$

25. A round balloon is being blown up so that its radius increases uniformly. Find the instantaneous rate of change of the surface area when the radius is 15 cm. Hint: What is the formula for the surface area of a sphere?

26. Given that  \( f(x) = \frac{\left|x + 1 \right|}{x + 1}, \; \; g(x) = \frac{x^2 – 1}{x – 1}  \), evaluate $$ \lim_{x \to -1^-} \left[ f(x) – g(x) \right] $$