Advertisements
Advertisements
Question
Examine the following function for continuity at the indicated point.
f(x) = `{((x^2 - 4)/(x-2) "," if x ≠ 2),(0 "," if x = 2):}` at x = 2
Advertisements
Solution
f(x) = `(x^2 - 4)/(x-2)`, also given that f(2) = 0
`"L"[f(x)]_(x=2) = lim_(x->2^-) f(x)`
[∵ x = 2 – h, where h → 0, x → 2]
`= lim_(h->0)` f(2 - "h") ..[∵ x = 2]
`= lim_(h->0) ((2 - "h")^2 - 4)/((2-"h") - 2)`
`= lim_(h->0) (4 + "h"^2 - 4"h" - 4)/(2 - "h" - 2)`
`= lim_(h->0) ("h"^2 - 4"h")/(-"h")`
`= lim_(h->0) ("h"("h - 4"))/(- "h")`
`= lim_(h->0)` h - 4
`= lim_(h->0) (0 - 4)/(-1)` = 4
But `"L"[f(x)]_(x=2)` f(2) = 0
∴ `"L"[f(x)]_(x=2) ne` f(2)
∴ The given function is not continuous at x = 2.
APPEARS IN
RELATED QUESTIONS
Evaluate the following:
\[\lim_{x->2} \frac{x^3 + 2}{x + 1}\]
Evaluate the following:
\[\lim_{x->∞} \frac{2x + 5}{x^2 + 3x + 9}\]
Evaluate the following:
`lim_(x->0) (sin^2 3x)/x^2`
If `lim_(x->a) (x^9 + "a"^9)/(x + "a") = lim_(x->3)` (x + 6), find the value of a.
If `lim_(x->2) (x^n - 2^n)/(x-2) = 448`, then find the least positive integer n.
Let f(x) = `("a"x + "b")/("x + 1")`, if `lim_(x->0) f(x) = 2` and `lim_(x->∞) f(x) = 1`, then show that f(-2) = 0
Show that the function f(x) = 2x - |x| is continuous at x = 0
For what value of x, f(x) = `(x+2)/(x-1)` is not continuous?
`"d"/"dx" (1/x)` is equal to:
If y = x and z = `1/x` then `"dy"/"dx"` =
