Limit

Limit of (x^n-a^n)/(x-a) as x approaches a: Formula, Proof

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The limit of (x^n-a^n)/(x-a) as x approaches a is equal to nan-1. This limit is denoted by limx→a (xn-an)/(x-a), so the limit formula of (xn-an)/(x-a) when x tends to a is given as follows. limx→a (xn-an)/(x-a) = n⋅an-1 Lets prove this limit formula. Proof of limx→a (xn-an)/(x-a) To prove limx→a (xn-an)/(x-a) = n⋅an-1 we will consider three different cases … Read more

Limit of (x^n-1)/(x-1) as x approaches 1

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The limit of (x^n-1)/(x-1) as x approaches 1 is equal to n, that is, limx→1 (xn-1)/(x-1) = n. This follows from the formula limx→a (xn-an)/(x-a) = n⋅an-1 Put a=1, so we get that limx→1 $\dfrac{x^n-1}{x-1}$ = n. Let us now prove this limit formula. What is the Limit of (x^n-1)/(x-1) Answer: limx→1 (xn-1)/(x-1) is equal … Read more

Limit of 1/x^2 as x approaches infinity

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The limit of 1/x^2 as x approaches infinity is equal to 0. As this limit is denoted by limx→∞ 1/x2, so the formula of the limit of 1/x2 is given as follows: limx→∞ 1/x2 = 0 What is the Limit of 1/x2 when x→∞ Answer: limx→∞ $\dfrac{1}{x^2}$ = 0. Explanation: As limx→∞ f(x)/g(x) is written as $\dfrac{\lim\limits_{x \to … Read more

Limit of x sin(1/x) as x approaches 0

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The limit of x sin(1/x) as x approaches 0 is equal to 0. This limit is denoted by limx→0 xsin(1/x). So the formula for the limit of x sin(1/x) when x tends to zero is as follows. limx→0 x sin(1/x) = 0. Let us now find the limit of xsin(1/x) using the Squeeze/Sandwich theorem. Proof … Read more

Limit of x^2 sin(1/x) as x approaches 0

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The limit of x^2 sin(1/x) as x approaches 0 is equal to 0, and it is denoted by limx→0 x2 sin(1/x) = 0. So the limit formula of x2 sin(1/x) when x tends to zero is given by limx→0 x2 sin(1/x) = 0. We will now find the limit of x2 sin(1/x) using the Sandwich/Squeeze … Read more

Limit of x^1/x as x approaches infinity

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The limit of x^1/x as x approaches infinity is equal to 1. This limit is denoted by limx→∞ x1/x, so the formula for the limit of x1/x when x tends to infinity is given by limx→∞ x1/x = 1. Let us now find the limit of x to the power 1/x when x tends to … Read more

Limit of (1-cosx)/x^2 as x approaches 0

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The limit of (1-cosx)/x^2 as x approaches 0 is equal to 1/2. That is, limx→0 (1-cosx)/x2 =1/2. This limit can be computed using the formula limx→0 $\dfrac{\sin x}{x}$ = 1 …(∗) Prove that limx→0 (1-cosx)/x2 =1/2 We have limx→0 $\dfrac{1-\cos x}{x^2}$ = limx→0 $\dfrac{2 \sin^2 \frac{x}{2}}{x^2}$ using the formula 1-cos2x=2sin2x. = limx→0 $\dfrac{2 \sin^2 \frac{x}{2}}{(\frac{x}{2})^2 … Read more

Limit of (1+1/x)^x as x approaches Infinity | Lim x→∞ (1+1/x)^x

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The limit of (1+1/x)^x as x approaches infinity is equal to e. Here we will discuss Lim x→∞ (1+(1/x))^x formula with proof. Note that Limx → ∞ (1+$\frac{1}{x}$)x = e. Limx→∞ (1+(1/x))x Formula with Proof The formula of limx→∞ (1+(1/x))x is given by limx → ∞ (1+(1/x))x =e. Explanation: Let y = limx→∞ (1+$\frac{1}{x}$)x. So … Read more

Limit of cosx/x as x approaches 0 | Lim x→0 cosx/x

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The limit of cosx/x as x approaches 0 does NOT exist. That is, limx→0 (cos x)/x is undefined. $\lim\limits_{x \to 0} \dfrac{\cos x}{x}$ = NOT exist. Let us now show that the limit of (cos x)/x when x tends to 0 does not exist. Limx→0 (cos x)/x Question: What is the limit of cosx/x when … Read more

Limit of cosx/x as x approaches infinity | Lim x→∞ cosx/x

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The limit of cosx/x as x approaches infinity is equal to 0. The formula of limx→∞ (cos x)/x is given below: $\lim\limits_{x \to \infty} \dfrac{\cos x}{x}=0$. Here we find the limit of (cos x)/x when x tends to infinity. Limx→∞ (cos x)/x = 0 Proof Question: What is the limit of cosx/x when x tends … Read more