Limits and Derivatives Class 11 Notes AP Inter 1st Year Maths Chapter 12

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Limits and Derivatives Class 11 Notes AP Inter 1st Year Maths 12th Lesson

→ The expected value of the function as dictated by the points to the left of a point defines the left-hand limit of the function at that point. Similarly, the right-hand limit.

→ Limit of a function at a point is the common value of the left and right-hand limits, if they coincide.

→ For a function f and a real number a, \({Lim}_{x \rightarrow a} f(x)\) and f(a) may not be the same (In fact, one may be defined and not the other one).

→ For functions f and g, the following holds:

→ The following are some of the standard limits:
\(\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}\)
\(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\)
\(\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=0\)

→ The derivative of a function f at a is defined by f'(a) = \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)

→ Derivative of a function f at any point x is defined by f'(x) = \(\frac{d f(x)}{d x}=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)

→ For functions u and y, the following holds:
(u ± v)’ = u’ ± v’
(uv)’ = u’v + uv’
\(\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^2}\) provided all are defined.

→ The following are some of the standard derivatives.
\(\frac{d}{d x}\left(x^n\right)=n x^{n-1}\)
\(\frac{\mathrm{d}}{\mathrm{dx}}\)(sin x) = cos x
\(\frac{\mathrm{d}}{\mathrm{dx}}\)(cos x) = -sin x

→ If y = f[g(x)] then \(\frac{d y}{d x}\) = f'(g(x)) . g(x)

Example Problems

Question 1.
Find the limits:

Solution:
The required limits are all limits of some polynomial functions.
Hence, the limits are the values of the function at the prescribed points.
We have

Question 2.
Find the limits:

Solution:
All the functions under consideration are rational functions.
Hence, we first evaluate these functions at the prescribed points.
If this is of the form \(\frac {0}{0}\), we try to rewrite the function, cancelling the factors which are causing the limit to be of the form \(\frac {0}{0}\).
(i) We have \(\lim _{x \rightarrow 1} \frac{x^2+1}{x+100}=\frac{1^2+1}{1+100}=\frac{2}{101}\)
(ii) Evaluating the function at 2, it is of the form \(\frac {0}{0}\).

(iii) Evaluating the function at 2, we get it in the form \(\frac {0}{0}\).

which is not defined.
(iv) Evaluating the function at 2, we get it of the form \(\frac {0}{0}\).

(v) First, we rewrite the function as a rational function.

Evaluating the function at 1, we get it in the form \(\frac {0}{0}\).

Question 3.
Evaluate:
(i) \(\lim _{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}\)
(ii) \(\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}\)
Solution:
(i) We have

(ii) Put y = 1 + x, so that y → 1 as x → 0.

Question 4.
Evaluate:
(i) \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 2 x}\)
(ii) \(\lim _{x \rightarrow 0} \frac{\tan x}{x}\)
Solution:

A general rule that needs to be kept in mind while evaluating limits is the following.
Say, given that the limit \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) exists and we want to evaluate this.
First, we check the value of f(a) and g(a).
If both are 0, then we see if we can get the factor that is causing the terms to vanish,
i.e., see if we can write f(x) = f₁(x) f₂(x) so that f₁(a) = 0 and f₂(a) ≠ 0.
Similarly, we write g(x) = g₁(x) g₂(x), where g₁(a) = 0 and g₂(a) ≠ 0.
Cancel out the common factors from f(x) and g(x) (if possible) and write \(\frac{f(x)}{g(x)} = \frac{p(x)}{q(x)}\), where q(x) ≠ 0.
Then \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{p(a)}{q(a)}\)

Question 5.
Compute \(\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}\)
Solution:
We have \(\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}=\lim _{3 x \rightarrow 0} \frac{e^{3 x}-1}{3 x} \cdot 3\)
= \(3\left(\lim _{y \rightarrow 0} \frac{e^y-1}{y}\right)\), where y = 3x
= 3 × 1
= 3

Question 6.
Evaluate \(\lim _{x \rightarrow 1} \frac{\log _e x}{x-1}\)
Solution:
Put x = 1 + h, then as x → 1 ⇒ h → 0.
Therefore, \(\lim _{x \rightarrow 1} \frac{\log _e x}{x-1}=\lim _{h \rightarrow 0} \frac{\log _e(1+h)}{h}\) = 1
(Since \(\lim _{x \rightarrow 0} \frac{\log _e(1+x)}{x}\) = 1)

Question 7.
Find the derivative at x = 2 of the function f(x) = 3x.
Solution:
We have

The derivative of the function 3x at x = 2 is 3.

Question 8.
Find the derivative of the function f(x) = 2x² + 3x – 5 at x = -1. Also prove that f'(0) + 3f'(-1) = 0.
Solution:
We first find the derivatives of f(x) at x = -1 and at x = 0.
We have


Clearly f'(0) + 3f'(-1) = 0.

Question 9.
Find the derivative of sin x at x = 0.
Solution:
Let f(x) = sin x. Then

Question 10.
Find the derivative of f(x) = 3 at x = 0 and at x = 3.
Solution:
Since the derivative measures the change in a function, intuitively, it is clear that the derivative of the constant function must be zero at every point.
This is indeed supported by the following computation.

Question 11.
Find the derivative of f(x) = 10x.
Solution:
Since f'(x) = \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
= \(\lim _{h \rightarrow 0} \frac{10(x+h)-10(x)}{h}\)
= \(\lim _{h \rightarrow 0} \frac{10 h}{h}\)
= \(\lim _{h \rightarrow 0}(10)\)
= 10

Question 12.
Find the derivative of f(x) = x².
Solution:

Question 13.
Find the derivative of the constant function f(x) = a for a fixed real number a.
Solution:

Question 14.
Find the derivative of f(x) = \(\frac {1}{x}\)
Solution:

Question 15.
Compute the derivatives of 6x¹⁰⁰ – x⁵⁵ + x.
Solution:
A direct application of the above theorem tells that the derivative of the above function is 600x⁹⁹ – 55x⁵⁴ + 1.

Question 16.
Find the derivative of f(x) = 1 + x + x² + x³ + ….. + x⁵⁰ at x = 1.
Solution:
A direct application of Theorem 6 tells that the derivative of the above function is 1 + 2x + 3x² + ….. + 50x⁴⁹.
At x = 1 the value of this function equals 1 + 2(1) + 3(1)² + …. + 50(1)⁴⁹ = 1 + 2 + 3 + ….. + 50
= \(\frac{(50)(51)}{2}\)
= 1275

Question 17.
Find the derivative of f(x) = \(\frac{x+1}{x}\)
Solution:
Clearly, this function is defined everywhere except at x = 0.
We use the quotient rule with u = x + 1 and v = x.
Hence u’ = 1 and v’ = 1.
Therefore

Question 18.
Compute the derivative of sin x.
Solution:

Question 19.
Compute the derivative of tan x.
Solution:
Let f(x) = tan x. Then

Question 20.
Compute the derivative of f(x) = sin²x.
Solution:
We use the Leibniz product rule to evaluate this.
\(\frac{d f(x)}{d x}=\frac{d}{d x}(\sin x \sin x)\)
= (sin x)’ sin x + sin x (sin x)’
= (cos x) sin x + sin x (cos x)
= 2 sin x cos x
= sin 2x

Question 21.
Find the derivative of f from the first principle, where f is given by
(i) f(x) = \(\frac{2 x+3}{x-2}\)
(ii) f(x) = x + \(\frac {1}{x}\)
Solution:
(i) Note that the function is not defined at x = 2.
But, we have

Again, note that the function f’ is also not defined at x = 2.
(ii) The function is not defined at x = 0.
But, we have

Again, note that the function f’ is not defined at x = 0.

Question 22.
Find the derivative of f(x) from the first principle, where f(x) is
(i) sin x + cos x
(ii) x sin x
Solution:

Question 23.
Compute the derivative of
(i) f(x) = sin 2x
(ii) g(x) = cot x
Solution:
(i) Recall the trigonometric formula sin 2x = 2 sin x cos x.
Thus \(\frac{d f(x)}{d x}=\frac{d}{d x}(2 \sin x \cos x)\)
= \(2 \frac{d}{d x}(\sin x \cos x)\)
= 2[(sin x)’ cos x + sin x (cos x)’]
= 2[(cos x) cos x + sin x (-sin x)]
= 2(cos²x – sin²x)
(ii) By definition, g(x) = cot x = \(\frac{\cos x}{\sin x}\)
We use the quotient rule on this function wherever it is defined.

Question 24.
Find the derivative of
(i) \(\frac{x^5-\cos x}{\sin x}\)
(ii) \(\frac{x+\cos x}{\tan x}\)
Solution:
(i) Let h(x) = \(\frac{x^5-\cos x}{\sin x}\)
We use the quotient rule on this function wherever it is defined.

(ii) We use quotient rule on the function \(\frac{x+\cos x}{\tan x}\) wherever it is defined.

Multiple Choice Questions

Question 1.
\(\lim _{x \rightarrow 1} \frac{x^2-1}{\mid x-1}\) = _______
(1) 0
(2) -2
(3) 2
(4) Does not exist
Answer:
(4) Does not exist

Question 2.
\(\lim _{x \rightarrow 10} \frac{1}{x} \cos ^{\prime} \frac{\left(1-x^2\right)}{1-x^2}\) = _______
(1) 0
(2) 1
(3) 2
(4) Does not exist
Answer:
(4) Does not exist

Question 3.
\(\lim _{x \rightarrow 0}\left(1+{tan}^2 x\right)^2\)
(1) \(\mathrm{e}^{\frac{1}{2}}\)
(2) e
(3) \(\mathrm{e}^{-\frac{1}{2}}\)
(4) 1
Answer:
(1) \(\mathrm{e}^{\frac{1}{2}}\)

Question 4.
If \(\frac{-\pi}{2}<\alpha<\frac{\pi}{2}\), then \(\lim _{x \rightarrow 2} \frac{(\cos \alpha)^x+(\sin \alpha)^x-1}{x-2}\)
(1) cos²α log cos α
(2) sin²α log sin α
(3) cos²α log cos α + sin²α log sin α
(4) 0
Answer:
(3) cos²α log cos α + sin²α log sin α

Question 5.
If ∆(x) = \(\left|\begin{array}{cc} e^x & -1 \\ \sin x-1 & 1 \end{array}\right|\), then \(\lim _{x \rightarrow 0} \frac{\Delta(x)}{x}\) = _______
(1) 0
(2) 1
(3) 2
(4) -1
Answer:
(3) 2

Question 6.
\(\lim _{x \rightarrow 0}\left(\frac{a^x+b^x}{2}\right)^{\frac{1}{x}}\) = _______
(1) \(\frac{1}{\sqrt{\mathrm{ab}}}\)
(2) \(\sqrt{a b}\)
(3) \(\frac {1}{ab}\)
(4) ab
Answer:
(2) \(\sqrt{a b}\)

Question 7.
\(\lim _{x \rightarrow 1} \frac{{tan} x-\sin x}{x^3}\)
(1) \(-\frac {1}{2}\)
(2) 0
(3) 2
(4) \(\frac {1}{2}\)
Answer:
(4) \(\frac {1}{2}\)

Question 8.
If a, b, d > 0 and ≠ 1 then \(\lim _{x \rightarrow 0} \frac{a^x \cdot b^x}{d^x-1}\)
(1) \(\log _d\left(\frac{a}{b}\right)\)
(2) \(\log _d\left(\frac{b}{a}\right)\)
(3) \(\log _{\left(\frac{a}{b}\right)} d\)
(4) \(\log _{\frac{a}{b}} d\)
Answer:
(1) \(\log _d\left(\frac{a}{b}\right)\)

Question 9.
\(\lim _{x \rightarrow 0}\left(\frac{1-\tan x}{1+\sin x}\right)^{\sin x}\)
(1) 1
(2) e^-1
(3) e^-2
(4) e
Answer:
(1) 1

Question 10.
\(\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-\sqrt{1-2 x}}{\sin ^1 2 x}\)
(1) -1
(2) 0
(3) 1
(4) \(\frac {1}{2}\)
Answer:
(3) 1

Question 11.
Assertion (A): \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) = 1
Reason (R): If \(\lim _{x \rightarrow a} f(x)\) = l, \(\lim _{x \rightarrow a} g(x)\) = m, then \(\lim _{x \rightarrow a} f(x) g(x)\) = lm
(1) Both A and R are true, and R is the correct explanation of A
(2) Both A and R are true, and R is not a correct explanation of A
(3) A is true, R is false
(4) A is false, R is true
Answer:
(2) Both A and R are true, and R is not a correct explanation of A

Question 12.
Assertion (A): \(\lim _{x \rightarrow 0} \frac{x}{x}\) = 1
Reason (R): f : R\{0} → R defined by f(x) = \(\frac{|\mathrm{x}|}{\mathrm{x}}\) has range {-1, 1}
(1) Both A and R are true, and R is the correct explanation of A
(2) Both A and R are true, and R is not a correct explanation of A
(3) A is true, R is false
(4) A is false, R is true
Answer:
(4) A is false, R is true

Question 13.
Let f(x) = x + [x], x ∈ R then f'(\(\frac {3}{2}\)) is equal to
(1) \(\frac {3}{2}\)
(2) 1
(3) 0
(4) \(\frac {5}{2}\)
Answer:
(2) 1

Question 14.
If f(x) = x cos x then f'(\(\frac{\pi}{2}\)) is equal to
(1) \(-\frac{\pi}{2}\)
(2) \(\frac{\pi}{2}\)
(3) \(\frac{\pi}{4}\)
(4) π
Answer:
(1) \(-\frac{\pi}{2}\)

Question 15.
If f(x) = 1 + x + x² + x³ + x⁴ + ….. + xⁿ, then f'(1) = _______
(1) n
(2) n + 1
(3) n(n + 1)
(4) \(\frac{n+n^2}{2}\)
Answer:
(4) \(\frac{n+n^2}{2}\)