Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
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.
Examine the following function for continuity at the indicated point.
f(x) = `{((x^2 - 9)/(x-3) "," if x ≠ 3),(6 "," if x = 3):}` at x = 3
Find the derivative of the following function from the first principle.
log(x + 1)
Find the derivative of the following function from the first principle.
ex
Evaluate: `lim_(x->1) ((2x - 3)(sqrtx - 1))/(2x^2 + x - 3)`
If y = log x then y2 =
