We are given
Andre rode his bike at a constant speed he rode 1 mile in 5 minutes
Firstly, we will find constant speed
In 5 minutes , distance travelled =1 miles
so, we get speed
Let's assume T is time in minutes
D is the distance in miles
we know that
so, we can plug value
and we get
so, we get
...............Answer
1. As you can tell from the function definition and plot, there's a discontinuity at x = -2. But in the limit from either side of x = -2, f(x) is approaching the value at the empty circle:
Basically, since x is approaching -2, we are talking about values of x such x ≠ 2. Then we can compute the limit by taking the expression from the definition of f(x) using that x ≠ 2.
2. f(x) is continuous at x = -1, so the limit can be computed directly again:
3. Using the same reasoning as in (1), the limit would be the value of f(x) at the empty circle in the graph. So
4. Your answer is correct; the limit doesn't exist because there is a jump discontinuity. f(x) approaches two different values depending on which direction x is approaching 2.
5. It's a bit difficult to see, but it looks like x is approaching 2 from above/from the right, in which case
When x approaches 2 from above, we assume x > 2. And according to the plot, we have f(x) = 0 whenever x > 2.
6. It should be rather clear from the plot that
because sin(x) + 3 is continuous at x = 0. On the other hand, the limit at infinity doesn't exist because sin(x) oscillates between -1 and 1 forever, never landing on a single finite value.
For 7-8, divide through each term by the largest power of x in the expression:
7. Divide through by x². Every remaining rational term will converge to 0.
8. Divide through by x² again:
9. Factorize the numerator and denominator. Then bearing in mind that "x is approaching 6" means x ≠ 6, we can cancel a factor of x - 6:
10. Factorize the numerator and simplify:
where the last equality holds because x is approaching +∞, so we can assume x ≠ -1. Then the limit is