Evaluate the following limits: limx→3x2-81x-3 - Mathematics

Evaluate the following limits:

`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`

Sum

`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`

Put `sqrt(x) - y`,

When `sqrt(x) -> 3`,

We have y → 3

`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3) = lim_(sqrt(x) -> 3) (((sqrt(x)^2))^2 - 3^4)/(sqrt(x) - 3)`

= `lim_(sqrt(x) -> 3) ((sqrt(x))^4 - 3^4)/(sqrt(x) - 3)`

= `lim_(y -> 3) (y^4 - 3^4)/(y - 3)`

`lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)`

= `4(3)^(4 -1)`

= 4 × 3³

`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)` = 4 × 27

= 108

shaalaa.com

Is there an error in this question or solution?

Chapter 9: Differential Calculus - Limits and Continuity - Exercise 9.2 [Page 102]

Chapter 9 Differential Calculus - Limits and Continuity
Exercise 9.2 | Q 3 | Page 102

Evaluate the following limit:

`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`

Evaluate the following limit:

`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`

Evaluate the following limit :

`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`

Evaluate the following limit :

`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`

Evaluate the following :

`lim_(x -> 0) [(sqrt(1 - cosx))/x]`

In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?

`lim_(x -> 2) f(x)` where `f(x) = {{:(4 - x",", x ≠ 2),(0",", x = 2):}`

Sketch the graph of f, then identify the values of x₀ for which `lim_(x -> x_0)` f(x) exists.

f(x) = `{{:(sin x",", x < 0),(1 - cos x",", 0 ≤ x ≤ pi),(cos x",", x > pi):}`

Evaluate the following limits:

`lim_(x -> 0) (tan 2x)/x`

Evaluate the following limits:

`lim_(x -> 0) (2^x - 3^x)/x`