In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why? limx→1sinπx - Mathematics

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 -> 1) sin pi x`

Graph

`lim_(x -> 1) sin pi x`

From the graph x = 1, the curve y = f(x) intersects the line x = 1 at x – axis.

∴ y = f(1) = 0

Hence `lim_(x -> 1) sin pix` = 0

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Concept of Limits

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Chapter 9: Differential Calculus - Limits and Continuity - Exercise 9.1 [Page 97]

Chapter 9 Differential Calculus - Limits and Continuity
Exercise 9.1 | Q 13 | Page 97

Evaluate the following limit :

`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`

In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.

`lim_(x -> -3) (3x + 2)` = – 7

Evaluate the following :

`lim_(x -> 0)[x/(|x| + x^2)]`

Evaluate the following :

Find the limit of the function, if it exists, at x = 1

f(x) = `{(7 - 4x, "for", x < 1),(x^2 + 2, "for", x ≥ 1):}`

Evaluate the following :

`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`

In problems 1 – 6, using the table estimate the value of the limit.

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

x	1.9	1.99	1.999	2.001	2.01	2.1
f(x)	0.344820	0.33444	0.33344	0.333222	0.33222	0.332258

In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (cos x - 1)/x`

x	– 0.1	– 0.01	– 0.001	0.0001	0.01	0.1
f(x)	0.04995	0.0049999	0.0004999	– 0.0004999	– 0.004999	– 0.04995

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 -> 1) (x^2 + 2)`

If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning